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This is to certify that the
dissertation entitled

Theoretical Study of Neutron Scattering

from Mott Insulators
presented by

Hyunju Chang

has been accepted towards fulfillment
of the requirements for

Ph.D. Physics

degree in

 

 

Tim; to XDNJW-iér.

T. A. Kap an andS . Mahanti

 

Major professor

Date August 25, 1995

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ABSTRACT

THEORETICAL STUDY OF NEUTRON SCATTERING FROM MOTT
INSULATORS

By

Hyunju Chang

Even though many theoretical works have been done in LagCuO4 and
YBa2Cu306, the insulating parents of the high-Tc superconductors, there is no sat-
isfactory theory to describe the low-temperature neutron scattering experiments
in these materials. This dissertation addresses our theoretical study of the neutron
scattering of the copper compounds, La2CuO4 , YBa2C11306 and SFzCUOgClz .

and the related nickel compounds. L32Nl04 , Ix'NiFg , and NiO. '

We develop a theory of the spin density which incorporates quantum spin
fluctuations (QSF) and covalence effects (COV) simultaneously. We found the
spin density, to a good approximations, to be a product of a site spin expectation
value which incorporates QSF and a form factor which takes COV into account.

The ordered moment is also defined from the spin density, and is found to be

affected by both the QSF and COV.

We have calculated the magnetic form factor with various theoretical tools and

the calculated form factors are compared with experiments.

As a simple procedure to obtain the form factor for LagNi04 , we follow the
model which was developed by Hubbard and Marshall. The Hubbard and Marshall

procedure failed to explain the observed large covalence in L32Nl04 .

We have carried out ab initio cluster calculations of the form factor in a solid us-
ing a cluster approximation based on quantum chemistry technique. We have used
restricted Hartree-Fock (RHF) and unrestricted Hartree-Fock (UHF) procedures
with correlation corrections via a restricted multi-configuration self-consistent-
field (MCSCF) approach to obtain the ground state cluster wave functions needed

in the neutron form factor calculations.

We find that the ab initio cluster calculations describe the experimental form
factor in KNiF3 and NiO extremely well. but fail badly in LagNiO4 . We ap-
plied the same method to the cuprate materials, La;CuO4 . YBagCu306 and
SF2CU02C12 . The calculated form factors for these cuprates turn out to agree
reasonably well with experiments. although there are other indications that the

degree of covalence is underestimated in our calculations.

To .113; .‘Uother

iii

ACKNOWLEDGEMENTS

I would like to express my deep gratitude to my thesis advisors Professor
T. A. Kaplan and Professor S. D. Mahanti for their physical insights to guide
me to understand physics during my graduate research years. I greatly appreciate
their constant support and the sharing of their enthusiasm for physics with me.
I also would like to greatly thank Professor J. F. Harrison for helping me to
understand ab initio quantum chemistry techniques. I have always considered

him as one of my advisors.

I am also grateful to Professors J. Cowen and P. Danielewicz for serving on
my guidance committee. I would like to thank Ms. Janet King for making my life

in this department so much simpler and easier.

I wish to thank Randy Rencsok and Dr. R. Boehm for helping me to solve the
problems encountered in using quantum chemistry computational codes. I also
would like to thank all my friends in the department for making .my life here more
easier and enjoyable. Especially I want to thank Dr. Hyangsuk Seong for being a

mentor in my graduate life.

I owe a great debt to my dear husband, Jaeyong Yee. This work could not
have been completed without his endless help and encouragement. I also hope
this work would be a little reward to my precious daughter. Lily. for the time she
had to spend without mommy. I also would like to thank my parents-imlaw for
their support and encouragement in many ways. Finally. I would like to dedicate
this thesis to my dear mother who has been always patient and supportive during

my studies.

iv

Financial support from the Natural Science Foundation and the Center for
Fundamental Materials Research of MSU. without which this thesis could not

have been done, is also greatly acknowledged.

Contents

Abstract .................................. i
Acknowledgments ............................ iv
1 Introduction 1

2 Theory of the spin density in an antiferromagnetic insulator in-

cluding covalence and quantum spin fluctuations 8
2.1 Introduction .............................. 8
2.2 Effective spin hamiltonian ...................... 10
2.3 Spin density .............................. 13
2.3.1 Formalism .......................... 13
2.3.2 Relation to the Hubbard-Marshall theory .......... 18

2.3.3 Correction term from H“) using the spin wave approximation 22

2.3.4 Estimation of the correction terms in spin density ..... 24

2.4 Ordered moment ........................... 25
2.5 Summary ............................... 26

3 Hubbard-Marshall model calculation 28

vi

3.1 Introduction .............................. 28
3.2 Hubbard-Marshall theory ....................... 29
3.3 Hubbard-Marshall model calculation for LagNi04 ......... 34
3.4 Discussion ............................... 38

4 Ab initio cluster calculation of neutron scattering form factor 41

4.1 Introduction .............................. 41
4.2 General theory of ab initio calculation methods .......... 42
4.2.1 Hartree-Fock(HF) Self-Consistent-Field (SCF) method . . 43
4.2.2 Beyond the Hartree-Fock approximation .......... 46
4.2.3 Natural orbitals ........................ 48

4.3 Theoretical form factor ........................ 50
4.4 Cluster and environment ....................... 53
4.4.1 Cluster ............................. 53
4.4.2 Point charge model ...................... 54
4.4.3 Effective core potential(ECP) ................ 54

5 Neutron scattering form factor of the nickel compounds 58
5.1 Introduction .............................. 58
5.2 KNiF3 ................................. 59
5.2.1 Experimental form factor ................... 60
5.2.2 (NiF6)“" cluster ........................ 63

vii

5.2.3 Environment

eeeeeeeeeeeeeeeeeeeeeeeee

5.2.4 Comparison with experiment .................

5.3 NiO ..................................
5.3.1 Cluster .............................
5.3.2 Environment .........................
5.3.3 Comparison with experiment .................

5.4 LagNiO4 ................................
5.4.1 Cluster .............................
5.4.2 Environment .........................
5.4.3 Comparison with experiment .................

5.5 Summary ...............................

Neutron scattering form factor of the cuprate compounds

6.1 Introduction ........................ ‘ ......
6.2 Cu2+ ion ................................
6.2.1 Ionic form factor .......................
6.3 YBagCU3Os ..............................
6.3.1 Cluster .............................
6.3.2 Environment .........................
6.3.3 Calculated form factor ....................
6.3.4 Comparison with experiment .................

viii

65

69

T4

76

76

79

79

83

83

84

88

90

93

98

6.4 L32CUO4

6.5

6.6

............................... 101
6.4.1 Cluster ............................. 102
6.4.2 Environment ......................... 103
6.4.3 Calculated form factor .................... 103
6.4.4 Comparison with experiment ................. 105
Sr2Cu02C12 .............................. 106
6.5.1 Cluster ............................. 108
6.5.2 Environment ......................... 108
6.5.3 Comparison with experiment ................ i . 108
Conclusions .............................. 110

Perturbation calculation of the contribution of the 4-spin hamil-

tonian 751(4) 113
Construction of natural orbitals in MCSCF . 117
3.1 Development of programs ...................... 118
Failure of the Mulliken Charge Population Analysis 122
Crystal field splitting in KNiF3 and N i0 125

ix

List of Tables

5.1

5.3

6.1

6.2

6.3

CI

D.1

Experimental values of < 52 > f((j‘M) in KNiF3 ..........
Madelung potential value at the origin of the cluster in Ix'NiFg . .

Madelung potential value at the origin of the cluster in NiO

Total energies (eV) of various calculations for Cu2+ ion ......
Comparison of total energies (eV) of clusters ............

Covalence factors obtained from the cluster calculations ......

Mulliken charge population values for different basis sets for Ni . .

lODq values for KNiF3 and MO ...................

64

..'69

87

96

110

123

126

List of Figures

3.1

3.3

3.4

4.1

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

6.1

Orbitals of a linear chain antiferromagnet. .............
Antiferromagnetically ordered N102 plane ..............

Schematic diagram of antiboding wave function in Ni02 plane . .

Comparison of Hubbard-Marshall model calculation with the ex-

periment for LagNiO4 . .......................
Effective core potentials for La3+ ..................

Crystal structure of KN iF3 .....................
Form factor Of'KNlF3 ........................
Crystal structure of NiO .......................
RHF Form Factor of NiO .......................
UHF Form Factor of NiO .......................
Crystal structure of L32N104 and LagCuO4 ............
Form factor of LagNiO4 with different ECP environment .....

Form factor of L32NIO4 compared with experiment ........

Form factor of Cu2+ ion with various calculation methods.

xi

31

36

39

*1

01

61

66

68

71

80

86

6.2

6.3

6.4

6.6

6.7

6.8

6.9

8.1

Comparison of form factor of free Cu“ ion. Cu2+ ion with point

charge distribution. and (Cu05)8‘ cluster form factor ......
Crystal structure of YBRzCUgOs ..................
Bilayer structure of Cu02 planes in YBagCu306 ..........
Form factor of YBagCugOa with various calculation methods.
Comparison of form factor of YBazCugOs with experiment . . . .
Form factor of LagCuO4 with various calculation methods.
Comparison of form factor of LagCuO4 with experiment. .....

Comparison of form factor of szCUOzClg with experiment.

Structure of program sets to construct NO’s ............

xii

94

97

99

104

107

109

121

Chapter 1

Introduction

Since the discovery of high-Tc superconductivity in La2_xSrrCuO4 and
YBagCu306+,, there have been many theoretical and experimental works
in La2CuO4 and YBagCugOs. the insulating parents of the high-TC
superconductors[chak90]. It is widely believed that the electronic structure of
these insulators is well understood [chak90]. However there are fundamen-
tal problems associated with the interpretation of the low-temperature neu-

tron scattering experiments in these insulating parents as discussed below

[kap191, kapl92, maha93].

Neutron Bragg scattering experiments found the ordered moment ( to be de-
fined later) to be in the range 0.60 ~ 0.66113 [yama87. tranSSB. bur188] ( this
value itself is still controversial as discussed in Chap.6). This result was inter-
preted [tranSSB] in terms of the spin 1/ 2 antiferromagnetic Heisenberg model for
spins on a square net ( which corresponds to C 11“” ions ( (19 configuration ) sitting
at the Cu sites on the C 110: planes). It was noted [tranSSB] that this result agreed
closely with the spin wave theory. and the large reduction from the nominal 1.1

pg ( taking 9 = 2.2 ) is due to quantum spin fluctuations [ande52]. The excellent

agreement with the spin wave theory was taken [traHSSB] to signify negligible con-
tribution of covalence to the reduction of the ordered moment ( the existence of
this covalent reduction had been noted by Hubbard and Marshall many years ago
[hubb65]). Moreover, recent Monte Carlo calculations [rege88. gr0589] agree with
the spin wave result, thereby reinforcing the idea of small covalent reduction of the
antiferromagnetic moment discussed above. The conclusion of negligible covalence
was recently questioned[kapl91] since these oxides are expected to have extremely
large covalence. The latter view was supported by a recent neutron diffraction ex-
periment in LagNiO... ( structurally same as LagCuO4 ). which apparently showed
a very large covalent contribution to the form factor with a covalent reduction
of the ordered moment by ~ 50% ! [wang91, wang92] The obvious intuitive no-
tion that both effects, spin fluctuations and covalence, should contribute to the

reduction of the ordered moment clearly leads to a contradiction in the cuprates.

Previous theories considered spin fluctuations without covalence. or covalence
without spin fluctuations, whereas these two should be treated simultaneously.
Here, in this work we consider these two effects simultaneously via the 1-band
Hubbard model in the %-filled narrow band regime. In particular. we calculate
the neutron scattering cross section. that is a measurable quantity experimentally.

allowing a unified theory of these tWo effects on the ordered moment[kap192].

As a consequence of this theory. we [kap192] were able to detect two errors in
the interpretation [tran88B] of the neutron scattering data. The interpretational
errors made by the experimentalists [tranSSB] are (i) they interpreted the site
spin incorrectly as the ordered moment rather than the site spin in the Heisenberg
model ground state and (ii) they used the form factor of I<2CuF4 to substitute

for that of the cuprate materials ; since K2CUF4 is a ferromagnet. its form factor

2

contains no momentum reduction which is expected in an antiferromagnet. In

order to resolve this situation, we have decided to calculate the form factor.

As a simple procedure to obtain the form factor. we follow the model calcula-
tion which was developed by Hubbard and Marshall [hubb65] for LagNiO4 . The
Hubbard and Marshall procedure failed to explain the data in LagNiO4 . This
motivated us to carry out a proper calculation of the form factor using an ab initio
cluster method based on quantum chemistry techniques. As far as we know. it
is the first time modern quantum chemistry methods have been used to calculate
the form factor in a solid. In order to justify the adequacy of the method. we have
carried out the cluster calculation for several Mott insulators including the insu-
lating parents of the high Tc superconductors. The ab inito cluster calculation
method describes the experimental form factor in KNIF3 and MO extremely well.
It is noted that this is the first satisfactory ab initio calculation of the form fac-
tor in KNiF3 and NiO. However the same cluster method fails badly in LagNiO4

[chan94, kap194]

We applied the same method to the cuprate materials. LagCuO4 . YdeCU305
and Sl'zCUOzClg , which we were originally interested in. The calculated form

factor for these cuprates turns out to agree reasonably with experiments.

The organization and the basic results of this thesis are as follows. First, in
Chapter 2, we describe our theory of the spin density which incorporates quantum
spin fluctuations and covalence effect simultaneously [kap191, kapl92]. We work
with the l-band Hubbard hamiltonian for the 1/2-filled regime and formulate the
expectation value of the spin density in the ground state using a perturbation

theory. The spin density is found to be a product of a site spin expectation value

I]

 

I it

 

and the form factor, plus some correction terms. We calculate the correction terms
and show they are negligible for the type of the materials being considered. Thus
the spin density is a simple product of the site spin and the form factor. The site
spin expectation value is calculated from the Heisenberg spin hamiltonian. which
includes quantum spin fluctuations but no covalence. and the form factor includes
the covalence effect but no quantum spin fluctuations. The ordered moment is
defined from the expectation value of the spin density, and is affected by both
quantum spin fluctuations and covalence. The expectation value the of site spin
can be estimated by the spin wave approximation[and652], which agrees with
recent Monte Carlo calculations[rege88, gr0389]. The reduction from its mean
field value is about 40% ( caused by quantum spin fluctuations only ). The rest of
this dissertation is concerned with calculating the form factor to investigate the

covalence effect.

In Chapter 3, we calculate the form factor for LagNiO4 using Hubbard-
Marshall(HM) theory [hubb65]. The experimental form factor of LagNi04 shows
about 50% reduction in the ordered moment which was claimed to be due to the
covalence in the form factor by Wang et. al.[wang91]. The HM analysis of the
ordered moment was first tried by Wang et. al. [wang91] to understand their
experiment. In the HM model, the wave function is a linear combination of the
ionic orbitals of one magnetic ion and neighboring ligands of the system. The
moment in the Ni2+ ion comes from the triple state where two d electrons are in
the a, state and have parallel spins. The choice of the same wave function for the
two parallel spin electrons ( made by Wang et. al. [wang91]) violates the Pauli
principle. Hence we decided to calculate the form factor with a physically mean-

ingful choice of the wave function in the e9 states in Ni“ . Our results disagreed

4

with experiment and moreover they were spoiling the excellent agreement that
had been obtained with that unjustified model by Wang et. al.‘s. Wang et. a1
[wang92] also came to a similar conclusion in their later work. We were there-
fore led to find more accurate methods. namely ab initio cluster methods based
on the Hartree-Fock (HF) self-consistent-field (SCF) technique. with correlation

corrections to calculate the form factor.

We introduce the ab initio cluster method in Chapter 4. First we review
the general concepts of the Hartree-Fock (HF) self-consistent-field (SCF) method.
Then we introduce the configuration interaction (CI) and multi-configuration self-
consistent-field (MCSCF) methods to include the correlation effects ( which go
beyond the HF method ). We derive expressions for the form factor within HF
or MCSCF approximations. We then describe our cluster approach and how we

treat the environment outside the cluster.

In Chapter 5, we describe the cluster calculations and results for the nickel
compounds. KNiF3 , NiO, and L32NIO4 [chan94. kap194]. First we apply our clus-
ter method to a rather simple antiferromagnet. KNIF3 by choosing the (N iF (5)4-
cluster. The ground state wave function of this cluster is calculated using the
restricted Hartree-Fock (RHF) and unrestricted Hartree-Fock (UHF) approxima-
tions. The calculated UHF form factor for KNiF3 agrees very well with the exper-
iment. Moreover we could estimate the quantum spin fluctuations quantitatively
by comparing our calculation with the experiment, where the absolute value of
the intensities of the Bragg peaks are available. The site spin expectation value,
obtained from the experiment using our theoretical form factor, agrees very well
with the value obtained using the spin wave approximation to the Heisenberg

model.

We then apply the RHF and UHF method to NiO taking (NiOs)‘°‘ as a
basic cluster. The calculated form factor agrees with the experiment [alpe61]
with some small corrections from orbital contribution. Unfortunately absolute
experimental values are not available in NiO. hence we could not determine the
site spin value. However the very complicated shape of the form factor agrees very
well with the experiment over the whole range of scattering vectors. (7 studied in
the experiment. giving further support to the accuracy of our procedure to obtain
the ground state wave function. We apply the same method to L&2N104 where
the experiment showed the large covalence. But our cluster method is in seriously

disagreement with the observed form factor in LagNi04 .

In Chapter 6. we address the calculation of the form factor in LagCuO4 .
SI20U02C12 , and YBa2Cu306 taking clusters. (Cqu)m' . (CuO4C12)8"and
(Cu05)8" . respectively. We show the results of RHF and UHF calculations.
We found that the calculated form factor shape agrees reasonably well with the
experiments. Especially, within a family of (i values characterized by increasing
only q,. the slope of f((f) vs ((11 is found to be nearly flat. It led us to conclude that
the spin density of the cuprate material is confined in the C L102 plane. We have
also improved our results by including correlation effect via multi-configuration
self-consistent field (MCSCF) procedure and introducing additional effective core
potentials (ECP) for environment. Unfortunately. the changes in the form factor

by these efforts turned out to be within the experimental errors.

Finally we discuss the difference between LagCuO4 and LagNiO4 . and try to
explain the failure of the cluster method in L32NIO4 and its sucess in KNIF3 ,
NiO, and other cuprate materials. We also discuss the covalence quantitatively

which we found in our cluster calculations and compare our results with an in-

6

dependent study on the covalence by Martin and Hay [mart93] using the similar
cluster methods for LagCuO4 . There is an indication of deficiency in our MCSCF
approach that the degree of covalence found in our MCSC F results was apprecia-
bly less than that of very extensive CI calculation by Martin and Hay. We suggest

that these additional correlation effect may explain the discrepancy in L212 NiO. .

Chapter 2

Theory of the spin density in an
antiferromagnetic insulator
including covalence and quantum
spin fluctuations

2. 1 Introduction

As we pointed out in Chap.1. earlier theoretical studies of the ordered magnetic
moment in antiferromagnetic (AF) insulators considered spin fluctuations without
covalence. or covalence without spin fluctuations. The theories of quantum or
zero-point spin fluctuations [kap192. rege88. grosSQ] were based on the Heisenberg
spin model. and therefore contained no covalence eflects on the antiferromagnetic
ordered moment. On the other hand. the theory of covalence effects on the ordered
moment [hubb65. akim76] were based on Hartree-Fock theory which contains no
quantum spin fluctuations. The effects. quantum spin fluctuations (QSF) and
covalence (COV) are conceptually different. Covalence is a local effect: it can
occur in one molecule with a magnetic ion bonded to diamagnetic ions; whereas

the existence of QSF in an antiferromagnet requires long range order and therefore

involves all the spins.

Here, in this chapter, in order to unify these two conceptually disparate effects,
we considered the l-band Hubbard model in the -;--filled narrow band regime. em-
ploying the mapping from a multi-band to the one-band model of Hybertson et.
al. [hybe90]. For the purpose of understanding ground and low-lying excited state
properties. the transition metal oxides with which we are concerned here can be
represented by a multi-band Hubbard hamiltonian. multiple bands corresponding
to d-orbitals of the transition metal ion and p—orbitals of the ligands. Further-
more. when p—orbitals are filled. the multi-band hamiltonian can be mapped into
a one-band (or two-band) Hubbard hamiltonian [hybe90] where the active bands
correspond to effective d-orbitals. For the cuprates. one has to deal with the singly

occupied eg orbital which results in a %-filled one-band Hubbard hamiltonian.

We obtain the expectation value of the spin density in the ground state of
this one-band Hamiltonian. Then we show that the Fourier transform of the spin
density can be expressed as a product of an expectation value of the site spin
incorporating QSF effect and a form factor including COV. with some additional
correction terms. The expectation value of the site spin is obtained with an
effective spin hamiltonian which is described below. We calculate the correction
terms and show that they are negligible in the %-filled narrow band regime. As
we shall show, the calculation of the spin density also allows a discussion’of the

ordered moment.

2.2 Effective spin hamiltonian

The one-band Hubbard Hamiltonian is given by

[\J
p-a
v

H = UZHHRQ + ZtgjafdaJ-a. ( .

ija

f . . . .
,0 creates a fermion in the one-particle state ww = w.(F)a., and n... 18

where a
the corresponding number operator. w.(f') = w(F — If.) is a Wannier function
associated with the magnetic ion of transition metal at site If.- and a, is the spin
state ( a = :l:1). The w.(1"')’s are real and form an orthonormal set. U is the
Coulomb repulsion between two on—site electrons (U > 0) and t.) is the hopping
parameter which is assumed to be non-zero, t., = t. for only nearest neighbor
pairs i and j. We consider the case where the hopping parameter t is very small

compared to U and the band is half filled, i.e. (the number of electrons) = (the

number of sites). In this spirit, we write Eq.(2.1) as

H=HO+V. (

to
[O
V

where Ho = UZ,n.-Tn.~1 and V = 2.1-, Marga]... treating V as the perturbation.

The Schrddinger equation of the system is written as
7'01! = E‘II. , (2-3)
We define P‘II with an effective hamiltonian He”. satisfying the relation of
H.,,Pv = EN, (fl-4)

where E and \II are the same as in Eq. (2.3). and P is the projection operator

which projects \II on to the ground state manifold of Ho. This manifold. called the
10

P-manifold. is characterized by one electron per each site and has dimensionalitv

(degeneracy) of 2N .

The ground state of Eq. (2.3) can be written as
‘1! = P‘Il + Q‘Il, (2.5)

where Q =1—P.

Rewrite Eq. (2.3) using Eq. (2.5)
H(P+Q)‘P = E(P+Q)‘II. (2.6)

and apply P on both sides of Eq. (2.6). then we obtain

P’H(P+Q)\Il = PE(P+Q)‘II
= EP‘II, (2.7)
because of P = 1 — Q.
If we apply Q on both sides of Eq. (2.6). then
Q'H(P+Q)\II = QE(P+Q)\II (2.8)

Using Q2 = Q and QP = 0 (which to follows from Q + P = 1). Eq. (2.8) becomes
Q [Q(H - E)QQ\II + ’HP\II] = 0. (2.9)

Then we obtain

1 .
Q‘Il = -Q(’H _ E)Q’HP\II. (2.10)

where W is the inverse of Q(H - E)Q, assumed to exist.

 

11

By plugging Eq. (2.10) into Eq. (2.7). we find

 

 

1
PHP‘P-PH HPW = EPW.
om — E)Q
Using P2 = P.
1
P P'HP - PH HP P‘I’ = EP‘II. 2.1
( Q(H - E>Q ) ‘ 1)

Then we obtain H.” by comparing Eq. (2.11) and Eq. (2.4). as

1
H. = P’HP — P’H ”HP. 2.12
If mH-EW ( )

 

We can write H — E = 7-1., - E, + V — 6E and expand W in terms of small
(V — 6E). Note that E, = 0. ’HOQ = UQ. and PHQ = PVQ. In the half-filled
narrow band regime, which is characterized by small 1H He]! Can be reduced to

a spin hamiltonian

.. .. t6
Heff = E Jng.‘ ' 51' + H“) + 0(a). (2.13)
‘1

_1t t _ t t ~z'
where Sf — 3(aita.1+ailaq). Sig — 517(ana.) — ailan), and b, = %(af,a.-T — aha”).
Here, the first term is the well-known Heisenberg Hamiltonian with J.)- = (tEJ/U)

and the second term . 71(4), is of order (tf/Ui’) and includes 4-spin operators.

These latter terms have been discussed in detail by Takahashi [taka77].

P111 is obtained from the lowest eigenfunction. <I>(31.sg, ...). of He), through

the relation
P‘Il = AII..w,.(F,.)<I>(sl,52, ...), (2.14)

where 1",. is the space coordinate of the n-th electron. s.- is the site spin coordinate,

and A is the antisymmetrizer operator.

1")

To the leading order in the hopping integral. QKII in Eq. (2.10). is given by
1
Q‘I’ = -UQ'HP‘II. (2.15)

Note that QHP = QVP.

2.3 Spin density

2.3. 1 Formalism

The Fourier transform of the spin density in the ground state ‘1! of the Hubbard

hamiltonian is

 

((P‘II +Qvns<om +Qv>1

(S(é’1)w = 1+ (leQv)

(2.16)

Since ‘1! is in the space of Slater determinants, we can use the spin density operator

3(5) in the form [kap191, kap192]

SUD = éngj-(flaaraaja (2.17)
= 30(q‘)+800(d'), (‘2-18)

where fU-(c'f) = fef‘f'Fw.(f")wJ-(F)df'. The first term of Eq. (2.18). so comes from
i = j and the second term, 300 comes from i # j terms. In the narrow band
regime, the overlap of the Wannier functions for i # j is small ; so we assume .j
to be order of t for i 75 j. In this spirit. we calculate the expectation value of the

spin density to 0(t2).
The leading term , of 0(t°), in Eq. (2.16) is
(8(5))0 = (PWlthDIPW)
= taxation. (219)
j

13

where f.,,((j) = [eff'fw(F)2dF. and the angular bracket means an average in the

P ‘11 space.

The contribution from .300. (P‘Illsomrflqul) in Eq. (2.16) is 0(t‘). It iden-
tically vanishes because 300 contains intersite terms which takes one out of the

P-manifold.

The terms of 0(t2) in Eq. (2.16) are

(3(4)). = (Q‘I’IsD(<i’)|Q‘I’>-(P‘PISD(®|P‘P>(Q\PIQ‘I’)

+<PWISOD(€)|Q‘I’) + (Q‘I’ISODUDIP‘I’). (2.20)

Here. the physically interesting quantity for us is the Fourier transform of the
spin density at if values corresponding to the Bragg peaks. So we are interested in
the values of q" = (Li. in Eq. (2.16). where (7A corresponds to the antiferromagnetic
Bragg wave vectors. The first two terms in Eq. (2.20). that is the contribution

from so. upon using Eq. (2.15), can be written as

(Q‘Pl:p(é1|Q\P) - (Wlsm wwcmw)
=f.(o(U— 2.): z e‘q’ita M,...s;.;a,.m.)

j (mn)aa’
—<. ;)(a1n,a..a;,,am..)), (2.21)

where the angular bracket means an average in the P‘Il space and the sum (m.n)
is over nearest neighbor pairs m.n and it is noted that m.n should be counted

differently from n, m.

The contribution from the third term in Eq. (2.20), son. is

t1

(P‘p1300(®le) = _U§ Z fmnl®<am charioaaiw'am0> ( '
(mn)aa’

lv
[(9
to

14

Since fmnfi): fnm( cf) and Z” ((11,, diamoafwoma ) = —Zaa.(alaamaaa;a,aw:),
Eq. (2.22) is identically zero. Similarly, the last term of Eq. (2.20). the another

contribution from .300. vanishes.

Now let’s carry out the summations on X)..." in Eq. (2.21) where
X1"... E (a amaaMSj ana,am,:) — (5;)(agaamalaomol). (2.23)

First. one can divide the summation Z) 2(mn) into two types of terms (1)

wherej '2 m orj = n and (2) wherej aé m and j aé n. i.e.

ZZ—VZXHZZ (.

1' (run) (in) J' (mn)¢i

[O
(Q
43

In the case (1),

2 Z Z x... = 2 X Z( (( aJUaMS ; aims...) — (5;)(ajaa..a;,,a,..)) (2.25)

(jn) do" (jn) 00’

The first term of Eq. (2.25) vanishes since 5 aJalP = 0 and the second term

3
J
d -.

2“. ajaanaaIWflJ-a: can be reduced to (% - 2S, - 5,.) after summation over 0.0’.

Thus Eq. (2.25) is reduced to

1 -' 2
222X...=-2ZZ(;—2< .-:.>)
J

(jn) aa’ "‘

(2.26)
where n is a particular nearest neighbor of j ( n can be any one of the nearest
neighbors), and Z is number of the nearest neighbors.

In the case (2), the summation over Xi"... can be written as

Z 2 ijm“ = Z 2 22(18):“ Inaa'watw'ama’)

2' (mm aw i <mn>¢j w'

-(5‘)(a moanaalalama > )

Z Z (<8;§m-§.>-<S;><§m-5.>). (22.27)

i (mn)¢j

15

Now let’s rewrite the summation 2] Emmy as ( see Eq. (2.24) ).

Z Z —~ZZ- '22 (2.28)

(mn)¢j J (mn) (Jn)

Using Eq. (2.28) in Eq. (2.27). we find that the first sum becomes

2(5)? 2 5"... §.)— (5;) Z (5"... 5”...) (2.29)

J (W!) (mn)
Noting that P‘I’ IS the ground state (eigenstate) )of :0... ,,)(
vanishes.
The second sum becomes
-2 2: [(575.- - 3‘.) — (52(53- - 2.)] . (2.30)

The first term of Eq. (2.30) can be written as

.. .. 1 _ fl
Ems - s.) = 2: [155.) + «(42»: — saw]. (2.31)

(in) (J?!)

. . 2 :1. _ "1: Z ‘1 "7y "v '1 ’w _ - my .
usmg the relations .5... - 5.. — 515,, + befi + 51.5}: and b bf — lb]. The 2nd
term of Eq. (2.31) vanishes because of symmetry on the summation of (jn) and

the first term becomes 2(1n)(1/4)(5§). Eq. (2.30) then reduces to

7221115371 ‘)(,--§ 3‘ )]. (2.32)
(M)
Using the relation (5;) = ~(Sj) for the nearest neighbor pair (jn) when there is
antiferromagnetic ordering, Eq. (2.32) becomes
1 .. ..
22: {-2- + 2(5. . 5.)] (5;). (2.33)
J. .

Combining (2.26) and (2.33), Eq. (2.20), the second leading term of Eq. (2.16)

becomes

2

<s(q*>>. = —27wa(<i')((-tj-2-)Z 61443;) (2.34)

16

where

7 21/4 —3(§.. - s...) (2.35)

Then we obtain a simple expression for the Fourier transform of the spin density

in Eq. (2.16), from Eq. (2.34) and Eq. (2.19). at Bragg peaks as.

(s (<11)=[1-27Z(-U2)(<Slfwr7) :F> («1) (2.36)

where F(cj) = Z]- efif'ffil'fll is the geometric structure factor that gives Bragg

peaks at the antiferromagnetic wave vector (74.

In Eq. (2.36), (5;) is the average site spin in the ground state of H.” . After

keeping terms of 0(t4) in ”He” [taka77], we obtain (5 j) to order of t2
t2
(5}) = (592.120 +655). (2.37)
where (52)”... is the magnitude of the sublattice spin per site in the ground state
of the Heisenberg Hamiltonian and 6 is a correction term coming from H”) in

Eq. (2.13). The details of estimating 5 will be discussed in Sec. (2.3.3).

Substituting the values of (5;) from Eq. (2.37) in Eq. (2.36). we obtain the

spin density (3(6)) as

(3(5))=(32)Ha.[1- -(27--Z Ugh )fw( (7)145) (‘3-38)

Eq. (2.38) is the central result of our calculations in this section. It gives the
Fourier transform of the spin density at the Bragg wave vectors as a product
of (52);“... the form factor of the magnetic site. and a moment reduction fac-
tor [l — (27Z — C(55)] arising from the hopping term of the one-band Hubbard

17

Hamiltonian. In the next section we will discuss how our results reduce to the

earlier theory of covalent reduction of the spin density given by Hubbard and

Marshall [hubb65].

2.3.2 Relation to the Hubbard-Marshall theory

To apply Eq. (2.38) to neutron Bragg scattering experiments, we need to know
explicitly the Wannier functions. w(77) associated with the magnetic ions. This

Wannier function can be expanded in the basis of Bloch functions.

1 _
F) = — 10', 2.39
m 2;: 1. ( 1
where N is number of sites. The Bloch function 111,-; can be expanded again in the

basis of unit cell functions 11(7" -— ff) which are not generally orthogonal. The point

is that the function u( (1") 13 available explicitly from atomic calculations.

(JP-F212 ”‘Ru (F— ii) (2.40)

C; can be determined by the normalization condition of the Bloch function.

(wrltl’t) = 1
= 12‘;- gze'fl(R'RI)/u(1’— ff)u(F— R’)dF, (2.41)
R!
where we assume we can neglect the overlap beyond nearest neighbors :
/u(F—R‘)u(F—I?)JF = 1 é—1?=o,
= A 172 — ii” = F .
= 0 otherwise . (2.42)

where r‘ is a vector connecting nearest-neighbor magnetic ions. From Eq. (2.41)

and Eq. (2.42), we obtain

ya = ( 1 .- )m. (2.43)
" 1+Azse‘k"

18

 

For small A , (/ C 12‘ becomes
1 3
y/C; = 1— 2.13ka + §A27EZ2 + . (2.44)
where ykZ 5 2,76”.

We can rewrite w(f") using MC; from Eq. (2.44) as

w(F) = i Z (1 _ 12.7.2 + 9.327322 + ...)e‘E'§11(F— R)

N - 2 8

Hi
= 11(5) +%;(-:A71Z) e“ RMF— Pt)
k.R

1 ..

+— (32127322) 8"“ IMF— R) + (2 15)
N m

The form factor in the Wannier function basis is then written. keeping terms

up to 0(A2) which we are interested in,

‘2 X 19' (7’1Z)/u(F— H)u(r‘)e‘5'é+"fidF

2 N -

. kfi

2 " ‘7 I I
+21% (71:2)(7HZ) /11(F— R’)u(1 — R)e""fq"'c R 1"“ 'dF

53.11)? I

3A2 2 ~_ “ iE-R+:d’-F .. 9
+2 X -— (71.2) u(r R)u(r‘)e dr. (...46)

8 N ER

Let’s consider the form factor fw((j) term by term in Eq. (2.46) at (7 = (7,4,

using ykZ E 2; eff”.

The first term of Eq. (2.46) becomes
1311:.) = [ewe-21;

= fu((I.A) (
19

tv
44
~l

The second term becomes

f1”(q = —A—- NEE/14F mae*(*+m+wwzF. (2.48)

USIDS % :1: e.£(F+fi) = 6F+R.01

fif’m‘) = —A:/u( 17+ T) u (fle‘q’dr.

-43: "F4“ fig")? (249)

where f( (q,r 1'"): fe‘q'u (r+ +;)u(f"— 9dr? Since f(cj’,7"°) = f(q", -—1"’) the summation

‘

on 1‘" can be written as lel(e““‘

e""'*'§)f(qf4.17). Then f12)(qf4) vanishes by

uh
+

:tiq'A-

”H

= :ti.

6

Similarly the third term becomes

f(3)((1')= _ZZ/u (+7? 7"“)u ("+F)e'q’d77 (2.50)

If 1' :,ér’ ,f(3’(q‘) vanishes by Eq. (2. 42). so we keep terms only when T— - 7’. Then

film.) reduces to

f13)(<7A) = - [:22 6““ Ff..(q.1)

2

One can obtain the fourth term by a similar way

f14>(q=13A22;/u(F+F — F)u(F)e‘q""dF. (2.52)

Keeping terms up to 0(A2) again, i.e. only keeping 1' = 1" terms, we obtain

1311'.) = iA’zma). (2.53)

From the above equations. the total fw((f,4) is given as

fwlia) = f11)(<f.4)+ f12)(<f.4) + f1?)(ii)+ HUM-:4)
A2

= f.(<i‘1)(1+Z-7). (2.59
Using Eq. (2.54) and keeping terms to order of t2. we can rewrite Eq. (2.38)
in the form

2 2

(sum = (5.1.1.3161 1+ 2% — 1221—51—5; Fm. (

(Q

.55)

Hubbard and Marshall[hubb65] considered a physical situation similar to the
C110; plane in LagCu04 and YBagCu306 compounds, where the paramagnetic
cations involving 3d electrons are surrounded by diamagnetic anions with filled

p-shells. They chose the unit cell function 11(7") as
"(7’) = C[610“) - [4213(7)], (‘3-56)
J

where d(1"') and pJ-(F) are atomic wave functions associated with cations and sur-
rounding anions respectively. C is a normalization constant and A is the d-p
covalence parameter. For our system, i.e. a C u02 plane, d(F) is a particular 3d
wave function of Cu2+ and pJ-(F')’s are the 2p wave functions of the surrounding
02‘.

We assume small A ( weak covalence ) and neglect overlap between anion or-

bitals and find that for arbitrary q’, the form factor f.(q) is obtained approximately
by

ma a f.(a(1— 2.12) — 2A 2111.11) — 2.1012(1)]

”221.1(7). (2.57)

A complete expression without any approximation is given in Chap. 3. In
Eq. (2.57). Z is the number of anions surrounding a cation ; for our system.
Z = 4. and fd(cj'), fp(q') and fdp)(q‘) are the Fourier transforms of (12(77), p307) and
d(F)pJ(f'). respectively. For (7: (Z... the contribution to the Bragg scattering from
the last term of Eq. (2.57) involving p2(f") can be seen to vanish. Thus only the
cl2 and the overlap (d — p) terms contribute to the form factor at magnetic Bragg

scattering.

If we take the square-bracketed term in Eq. (2.55) as 1. and (5;)Hm as the
mean field value. i.e. (52)”... = 0.5. and plug Eq. (2.57) into Eq. (2.55).
we obtain the Hubbard and Marshall result. described in Chap. 3. However,
Eq. (2.55) clearly has both the effects of quantum spin fluctuations ( in (32)”...
) and covalence ( in f..(q‘) ), with some additional correction terms. According
to our formalism, we should consider both these effects simultaneously and take
account of the additional terms of 0(t2). In the next sections. Sec.(2.3.3) and
Sec.(2.3.4). we will show that the correction terms are negligible for the systems
under study and the major sources of reduction of the spin density come from

quantum spin fluctuations and the (p — d) covalence effect.

2.3.3 Correction term from 71(4) using the spin wave ap-
proximation

To calculate (3;) in Eq. (2.37). we take H.” up to 0((—%) in Eq. (2.13). The
term of Hoff is well discussed by Takahashi[taka77]. We also follow Takahashi’s
method [taka77] and use the spin wave approximation. which is an expansion in
inverse powers of the spin quantum number 5. Our derivation of H“) agrees

with Takahashi’s result and we calculate (5;) using perturbation method with

(O
[Q

the ground state of ”H.” including H“).

From Eq. (2.12), the effective Hamiltonian can be written as
H,” = Ho + H“). (2.58)

where H0 is the Heisenberg Hamiltonian. Hm... as given by

12 1 ~ ~
=__ __2m.a , 2
H. U(§)(2 S ) (59)
and
H14) = Ug—PVQVPVQVP - al.—sPVQVQVQVP- (‘3-60)

Then we treat H“) as perturbation to ’Ho. The perturbation method is applicable
because H“) is C(55) whereas H0 is O(%) in the narrow band regime (t < U).
H“) including 4-spin operators can be written in the form shown in Eq. (A.1) of

Appendix A.

The expectation value of the site spin operator. 5;. is given by
(5;) = (\IIISfl‘II).. (2.61)

By the perturbation method, writing ‘11 = ‘11., + 641. we obtain (5;) up to the first

correction
(5}) = (\I’OISH‘I’O) + 2(WOISII5‘P) (‘2-62)

Using the spin wave approximation described in Appendix A. one finds
(IIIOIS;I\IIO) = (3;)381, = 0.30362 and 2(‘IIOISJZI6‘II) = 12839-55. If we rewrite
Eq. (2.62) as

z . 1.2839 12 9
(5,) - (3.1.... [1+ 0.30362 (55)] , (-.63)

 

then we obtain the correction term 6 = 4.23 in Eq. (2.37).

23

2.3.4 Estimation of the correction terms in spin density

In this section. we analyze the correction terms in the square-bracket of Eq. (2.55).
First. we easily find 7 in Eq. (2.35) from the known ground state energy ( in unit of
J) [gr0589. rege88], because (8,. 5"...) is proportional to the ground state eigenvalue
of the Heisenberg Hamiltonian. The value of 7 is 1.26. It is interesting to note
that '7 is greater than its mean field value of 1. In the Neel state ( mean field ),
only the z-components of spins are antiparallel while the x- and y- components
are not correlated. On the other hand, in the exact ground state of ”He”, the
spin fluctuations can cause an increase in the probability of antiparallelism of all
the three components of the spin thus providing a plausible explanation why 7 is

greater than 1.
For 5 in Eq. (2.55), we obtain 6 = 4.23 from the previous section.

The next correction term we are concerned with is value A which is defined
in Sec. (2.3.2). If we choose the unit cell function in the form of Eq. (2.56). we
find A = 2A5 — A2, where 5 = fp(f")d(f')d3r > 0. From Wang et. al.’s Hubbard-
Marshall model calculation [wang91], A = 0.35 and S = 0.175. for these values A
accidently vanishes. This suggests A should be extremely small. We take (t/ U)
in Eq. (2.55). as (fi) = 0?? = 0.08. from Hybertsen et. al. [hybe90] who mapped

the three-band Hubbard model onto a one-band model.

Using the above values of 7,5. etc., we find the square-bracketed term in
Eq. (2.55) is about 0.96. Thus our results suggest that these corrections can be
safely neglected compared to the well-known reduction ( about 40% of (5.) from
its mean field value) due to spin fluctuations. We expect similar or smaller values

for other materials we are interested in. So. we obtain a simple form for Eq. (2.55)

24

( we drop the subscript u for simplicity from now on ). as

9(a)) = (53)”...f(9‘.4), (2.64)

where

f(‘f:4) = fd(é'A)(1- 2A2) - QAZIprJé'A) - fdp,(0)fd(<ii1)]. (2-65)

when we assume small A and neglect the overlap between anions.

Here we define f,.(q‘) from Eq. (2.57) without the last terms. i.e. without p2

terms,

[4(6) = fdfffxl — 2A2) " 2A21fdp,(® —' fdp,(0)fd(q‘)1- (2-66)

Note that [443.) = f(q’,() for non-zero (7A.

2.4 Ordered moment

We are interested in the ordered moment of these AF insulators including the
effects of both spin fluctuations and covalence. The ordered moment. MO...) is
determined from the spin density (3(a)) times 9/13 by extrapolation i.e. by
putting it = 0 in Eq. (2.65). When qt. = o. the 2nd term in Eq. (2.65) vanishes,

and the ordered moment is

Mord = g”B(Sz>Heis(l ‘- 2A2) (267)

This agrees with Hubbard and Marshall who argued that the ordered moment (

they called it the effective moment ) will be reduced from the Heisenberg value

“1362);“, ( which they obtained in the mean field theory ) by the covalence

reduction factor, (1 - Z A2).

In general, without the approximation of small A and S. the covalence reduc-
tion factor. Rea”, can be defined as

w

Rcov = f(0) a

(2.68)

where f(0) is normalized to 1 from the definition of f((f), but [4(0) is less than
1 because f,4((f) does not contain the contribution from the pi terms associated
with the ligands which identically vanishes at (7 = q] due to antiferromagnetic

ordering (corresponding to the ordering of nearest—neighbor magnetic ions).

The ordered moment, film; is in general given in the form
Mord = g#B(Sz>Heisz(O)- ('3-69)
2.5 Summary

In this chapter, we showed how the two concepts, namely covalence (COV) and
quantum spin fluctuations (QSF), are unified in the spin density in a simple form
given in Eq. (2.64). We also showed that other correction terms in (3(i4)) up
to 0(t2/U2) are negligible and the main contributions to the reduction of (3%”)

from the nominteracting AF ordered spins are QSF in (53);“), and COV in f,4(cf).
The experimentally measured quantity is the magnetic moment density m(q"A)

which is

m(qA) = 9#B(3(<ili))

ng(Sz>Heisf(i4)' ( °

lv
*1
C)
V

Here only f (i4) has i—dependence and the other factors being constants. If we
calculate f((TA) and compare it with the experimental m(iZ4), we can directly
determine (52);“, from the experiment. The rest of this thesis is devoted to the
calculation of the form factor “(111)- The form factor carries information about not
only COV but also QSF if the absolute experimental values of m(('1f4) are available.
The shape of the form factor reflects the spin distribution in real space. Therefore
the form factor calculation is critical to an understanding of the magnetism in the

antiferromagnetical systems in which we are interested this thesis.

[0
*1

Chapter 3

Hubbard-Marshall model

calculation

3.1 Introduction

The effect of covalence in the neutron scattering form factor was first discussed by
Hubbard and Marshall [hubb65]. Hubbard-Marshall (HM) theory takes the wave
function for a system as a linear combination of atomic orbitals, thus allowing
charge and spin transfer from the paramagnetic ion to the nearby diamagnetic
ions. The covalence is a measure of the degree of this spin transfer. Akimitsu and
Ito [akim76] calculated the neutron scattering form factor for K2CUF4 following

HM theory and their calculated result agreed very well with the experiment.

Recently a HM model calculation for LagNiO4 was reported by Wang et. al.
[wang91] and their calculation for the form factor also agreed very well with the
experiment. These authors claimed that the large reduction of the ordered mo-
ment in their experiment was due to the covalence effect (see Chapter 2 regarding
a discussion of the effect of covalence on the antiferromagnetic ordered moment).
However the wave function for the system they used was rather unphysical. They

took one spherically symmetric d orbital for the two parallel electrons instead of

28

2 different orbitals of eg symmetry as is appropriate for the Ni2+ ion in a cu-
bic field. This violates the Pauli principle that prohibits two parallel electrons
from being in the same spatial orbital. We have calculated the form factor for
L32NIO4 taking 2 eg orbitals, d,2_y2 and d3z2_,.2, and compared our result with
the experiment [chan91]. Wang et. al. also reported a similar calculation in their
later paper [wang92]. However, this physically meaningful choice for the wave
function turned out to spoil the excellent agreement with experiment obtained
with one spherical orbital. Considering the sucess of the HM model calculation in
K2CuF4 , the failure of the similar method for LagNiOl is somewhat surprising,
since LagNiO, has the same structure as I<2CUF4 . This failure contributed our

motivation to initiate ab initio calculations of the neutron scattering form factor.

In this chapter, we introduce HM theory for covalence in the neutron scattering
form factor, and then describe our calculation for LagNiO4 within the HM theory,
and compare with experiment. It should be noted that the concept of covalence in
the form factor within HM theory is still valid in our ab initio calculation discussed
in the later chapters because it also starts from a linear combination of atomic

orbitals called basis functions.

3.2 Hubbard-Marshall theory

Hubbard and Mashall considered the transition metal compounds, which consists
of a. transition metal ion with unpaired electrons in d orbitals, and ligand ions
of either F ' or 02", each of which has the configuration 1322322p6. In this
system, a transfer of electrons is allowed from the ligand to the magnetic ion. This

transfer is mainly p—electron transfer to the singly occupied (id-orbitals, creating

‘29

an unpaired spin density on the ligand ions. Because of Pauli principle, the
spin of the transferred electrons must be opposite to that of the electron in the
singly occupied 3d state. Hence the spin density, created in the ligand orbitals, is
parallel to that of the 3d orbital. The electron transfer from the ligand ion to the
magnetic ion results in the spin transfer from the magnetic ion to the ligand ions.
In this instance the moment of the magnetic ion is decreased as a consequence
of the covalence. When the neighboring magnetic ions are antiferromagnetically
ordered, the net spin density created in the intervening ligand ion can be cancelled,
depending on the symmetry of the crystal structure. For ferromagnets, of course,

there is no cancellation of the spin density created in the ligand ions.

The cross section for magnetic Bragg scattering is proportional to the square
of the magnetic form factor associated with the magnetic ions and the form factor
reflects the spatial distribution of spin density. In an antiferromagnet, when a
ligand is shared equally by two antiferromagnetically ordered magnetic ions, the
form factor is reduced from a purely ionic value, because a certain amount of the
spin density, which is transferred to the ligands and is cancelled there, does not

contribute to the peak intensity.

For an illustration of the covalence effect, let us consider an antiferromagnetic
linear chain consisting of alternate single magnetic and ligand orbitals shown
in Fig. 3.1. Each d orbital contains just one electron, and these are taken to
be antiferromagnetically ordered in l-dimension, while each p orbital contains 2

electrons. In Fig. 3.1, the antibonding orbital associated with the ion M is

2120’) = N [4(7‘”) - AW”) + Ap'ml) (3-1)

30

p" d" p d P,

924: 020

L" M" L M L'

Figure 3.1: Orbitals 'of a linear chain antiferromagnet. M and M” are the magnetic
ions with antiparallel spins.

31

where the normalization constant N is given as
N"2 = 1 — 4A5 + 2.42 (3.2)

and S is the magnitude of the overlap integral between p and d orbitals.

The spin density associated with M is

D(F) = «WV

z N? (as? - 2Ad(-F){p(r*> - M} + A2{P(F)2 + pins] (3.3)
where we have neglected the overlap between p and p’.

The first term of Eq. (3.3) involves d(f')2 and is rather confined to the parent
magnetic ion M; the second term is an overlap density and is also confined to the
immediate vicinity of M; but the third term gives a density entirely on the ligands
and is equally distant from the other magnetic ion, 31”. The spin density given
by M” is similar to that by M, but is of opposite sign in an antiferromagnet. In
particular, we notice that for the moment density 1412;)(1?)2 due to M, there is an

equal and opposite contribution from M ”.

The total spin density in a crystal, associated with a N-magnetic ion system,

can be written as
D = figanDW-nlf), (3.4)

where 17’. is the nearest neighbor vector connecting M and AI”, and an is the
site spin corresponding to antiferromagnetic ordering. It is noted that for an
antiferromagnetic system on = 6‘93“: for an antiferromagnetic wave vector (Z4.

In (3.4), the contribution A’p(f")2 due to M is cancelled by the --Azp(i’)2 due to

M ” ; in fact all the M?)2 terms cancel in the same way.

32

The form factor, which is the Fourier transform of the spin density of Eq. (3.4),

can be written as

f(<i‘) = Yb-Ze‘f""R/D(F—n§)e‘f”df‘

At (1': (7A, Eq. (3.5) becomes
flit) = / D(fle"°’A'*’dF. (3.6)
It is noted that the contributions from p(1"')2 and p’(1"')2 terms in Eq. (3.3) to the
form factor in Eq. (3.6) vanish because of fp’(r"')ze“7""dF = e‘fA'pr(F)2e“7-4'F and
6‘74": = -1. Thus there is no contribution of p(F)2 or p’(i’)2 term to the form
factor. This reflects the above explanation that the spin density of ligand site

associated with M, to the right of M, cancelled by the corresponding contribution

associated with 1W”.

When A and S are small, we can rewrite D(F) in Eq. (3.3) using Eq. (3.2) up

to second order,
W) m d(F)2(1+ 4A5 — 2A2) — 2Ad(F){p(r”) -— p’m} (3.7)
without the third term of Eq. (3.3).
After rearranging Eq. (3.7) in the form
W) = d(r‘)’(1 — 2A2) + 2.4[25d(r‘)2 - d(r‘)p(r“) + dmp'm], (3.8)

it is noted that when we integrate D(F) in Eq. (3.8), the second term ( the terms
inside the square bracket ) vanishes. This results in the reduction of the ordered

moment by the factor (1 — 2A2). When we calculate the form factor taking the
33

Fourier transform of Eq. (3.8), the second term of Eq. (3.8) contributes to the
form factor as a positive value (although small) at (i aé 0, that passes through a
maximum as q" increases. This makes the form factor appear flatter in the small

ltfl region compared to the pure ionic form factor. which is given by the Fourier

transfer of d(F)2.

3.3 Hubbard-Marshall model calculation for
L32Ni04

The experimental form factor of LagNiO4 [wang91] showed a plateau in the small
)q‘l region, as expected within HM theory described in the previous section. There-
fore we applied Hubbard-Mashall theory to LagNiO4 taking two different (1 or-
bitals. These were chosen to be of 6,, symmetry, in accordance with expectations
from crystal field theory. We took the basic cluster consisting of one magnetic
Cu“ ion and 4 ligand 02‘ ions in the MO: plane where Ni2+ ordered antiferro-

magnetically as shown in Fig. 3.2.

In order to construct a wave function of the type given in Eq. (3.1), we used
Hartree-Fock atomic orbitals for both Cu2+ and 0‘ ions, which were obtained
from the restricted Hartree—Fock (RHF) method using COLUMBUS code [colu88].
For 19(17) atomic orbitals of 02‘ , we scaled the 0' atomic orbital in the If] space,
to give the same overlap integral value, 5 = 0.175, for air: and pat of Wang et. al.
[wang91]. Since the ground state of Ni2+ is spin triplet state, we considered two

cg orbitals, d,2_y2 and d322,,2, for each magnetic ion.
The spin density D(7"') is given as

W) = ¢r2—y2(7")2+¢322—r2(77)2- (3.9)

34

 

 

 

9

Ni2+

 

O“).

02"

 

 

 

 

{} f}

f}

Figure 3.2: Antiferromagnetically ordered NIOz plane. The arrows represent spin
up or down.

35

 

Figure 3.3: Schematic diagram of the antibonding wave function associated with
the d,2-,z orbital in NiOg plane

36

The wave function 1129-330") and $322,441?) can be written in a general form

1,9147?) = N1: (did?) -' AkZijfFl) . (3.10)

where k = l and k = 2 represent d,2_y2 and d3,2_,2 respectively and the j index
corresponds to the j-th ligand. The orientations of the pkj’s are chosen to produce

antibonding orbitals with dk, with positive covalence parameter A), as shown in

Fig. 3.3.

The normalization constant Nk, when the nearest neighbor p— p’ overlap, SW),

is included, is given as
N;2 =1-2ZA,.S+ ZAfi(1+25,,,,.) (3.11)

where Z = 4, the number of neighboring ligands.

The form factor associated with 11)), is

ft(<i‘) = NE/[dflfl—‘zAermphvi

+Ai ZPdelijiFll e‘q‘fllr’. (3.12)

i,j

For (i: i4,
fk(‘fA) = fd(‘fA) ‘ 2Akfdpl‘fA) + QAfcfpp’li-lla (3-13l
where

me.) = N2 [moe‘q‘r’da
that.) = N: jzdt(r”)ptj(7’)e‘f""dfi
J

frp'(élt) = N3 / Spitfireoae‘q‘flda (3.14)
(if)

37

The 2(0) in Eq. (3.14) means summation over the pairs of nearest neighbors, be-
cause the i = j case of the last term in Eq. (3.12) is cancelled by antiferromagnetic

ordering, and because the 2nd-neighbor p — p overlap is very small.

The total form factor is written as

flit) = kaUi'A) (3-15)
1:

Note that fk(0) = 1 when we use (3.12) ( whereas fk(0) < 1 if we use (3.13) ).
In Fig. 3.4, this flak) is compared with the experiment [wang91] by choosing the

covalence parameters, A1 and Ag, to give the best fit to the experiment.

3.4 Discussion

In Fig. 3.4, our more realistic model calculation shows worse agreement with the
experiment than Wang et. al.’s model calculation [wang91], even though we chose
physically meaningful wave functions. Our choice of A1 and .42 turned out to be
rather big because the fitting procedure tried to reproduce the large reduction
of the moment (by about 50%) which was seen in the experiment. Considering
the fact that HM theory starts from the ionic limit, which presumes small A’s,

LagNiO4 might be too covalent to be treated within HM theory.

When we compare our HM model calculation for LagNi04 with a similar cal-
culation in K2CUF4 where the HM model calculation agreed well with experiment
[akim76], we have to take account of the difference between Cu2+ and Ni2+ . The
ground state of Cu2+ has a singly occupied draw: orbital. Hence the spin den-
sity distribution is confined in the Cqu plane and spin transfer mainly occurs
within the plane. But both draw: and (1332-,2 are singly occupied for N i2+ . The

spin distribution of (1322-,2 is 3-dimensional, so the model we have considered here

38

 

2.0 IIIIIIITIIIIIIIIIIIIIITI

" q

f Experiment -
X Wang et.a1. model :
1-5 B Our HM model —«V

Cl

{:3 $ng

”fit.

X

f(qi)

0.5

IIIIIIIIIIIIITIjII

‘

 

 

 

 

 

oollilllllLllLll llllllll

0.0 0.1 0.2 93.3 0.4 0.5
sine/A (A-l)

Figure 3.4: Comparison of Hubbard-Marshall model calculation with the experi-
ment for LagNiO4 [wang91]. The square symbols are our HM model calculation
with 2 e, orbitals and the diagonal cross symbols are Wang. et. al’s model calcu-
lation [wang91] using spherical d orbitals ( see the text for details ). It is noted
that the total form factor is normalized to 2 in this case. The covalence parame-
ters, A1 and A2, in our form factor calculation, were chosen to be 0.40 and 0.30
respectively by least square fit to the experiment.

39

where there are one Ni2+ ion and 4 ligand 02' ions in the MO; plane, might not
be appropriate for LazNiO4 . We have to allow for out of plane spin transfer for
the Ni compounds. It is noted that the out of plane spin transfer is not cancelled
because the out of plane oxygen is not shared by two antiferromagnetically or-
dered Ni“ ’s. There is another possibility where there is a spin transfer to the 2s

orbitals of the ligands, which were not included in our model calculation.

In order to overcome this deficiency of our form factor model calculations,
we have carried out ab initio cluster calculations of form factors that start from
molecular orbitals which are linear combinations of basis functions including all
the possible orbitals in the ligands as well as in the magnetic ion. In the cluster we
take into account the out of plane oxygens also. The next three chapters are about

the ab initio cluster calculations of the form factor of transition metal compounds

including L32NIO4 .

40

Chapter 4

Ab initio cluster calculation of
neutron scattering form factor

4.1 Introduction

In order to resolve the puzzle about the degree of covalence in the ordered moment
of the cuprate materials discussed in Chap.1, we had to find a more accurate theory
to calculate the form factor. The Hubbard-Marshall model does not seem to be a
good model to get correct covalence because it failed to give the form factor for

L32Nl04 correctly as discussed in Chap.[3].

A band calculation of the form factor of L32NIO4 using the local-spin-density
approximation (LSDA) was reported by Wang et. al. [wang92]. However their
calculated magnetic moment was found to be far too small, 0.45 #3 for each Ni2+ ,
which is much smaller than the observed value, namely about 1.0 pg. When they
compared the shape of the form factor with experiment, the agreement was rather
poor[wang92]. The case of L32CUO4 was found to be even worse. LSDA did not
give the observed antiferromagnetic ordering for La2Cu04 [pick89]; i.e. it gave
zero for the ordered moment. In SrgCu02C12 , Wang et. al. [wang90] using LSDA
also got a very small moment and the agreement of the shape of the form factor

41

with experiment was as poor as in LagNiO4 . Poor agreement between LSDA (
or LDA ) results with experiment suggests that these approximations do not give
a correct description of the ground state wave function of the system because of

the strong intra-atomic correlation in LSDA .

At this point we decided to use quantum chemical ab initio methods to
calculate the ground state wave function and the neutron form factor. There
were some ab initio cluster calculations for LagCuO4 and YBa2Ct1306 reported
[mart91, sula90] in the literature. But none of them studied the magnetic form

factor.

In this chapter, I give a brief review of the quantum chemical ab initio methods:
the Hartree-Fock (HF) approximation, and multi-configuration self-consistent-
field (MCSCF) and configuration interaction (CI) approaches beyond HF. I de-
scribe how we calculate the form factor from the wave function obtained by ab
initio methods. Then I address how we choose the cluster and how we treat the

environment outside thetcluster.

4.2 General theory of ab initio calculation meth-
ods

In this section, I review various ab initio methods to obtain the ground state wave
function of a cluster using quantum chemistry technique. This review is based

on Prof. J. F. Harrison’s lecture notes and some reference books [levi91, mcwe89,

lowe78].

In general an ab initio calculation via the variation method rigorously involves

the following steps.

(1) Write down the hamiltonian operator 'H for the system.

(2) Select some mathematical functional form 111 as a trial wave function with

variable parameters.
(3) Minimize E = (ilil’fitldi) with respect to the parameters.

The term ab initio is used to describe calculations in which the three steps
listed above are all explicitly performed. In this section, we describe a certain

kind of ab initio calculation called the self-consistent-field (SCF) method.

4.2.1 Hartree-Fock(HF) Self-Consistent-Field (SCF)
method

In the non-relativistic limit, the electronic hamiltonian, H , for an N electron

system can be written as follows, ( We use atomic units.)

H=Zfi+ZZ§u (4-1)

j>i i
where
1 2,,
f. = -.—V?- —. (42)
2 a ria
, 1
r.)

where theindices i, 1' refer to the electrons and 0: refers to the nuclei of the system.
In atomic units, the electron charge, e, is 1 and the Bohr radius a0 is. 1, the

atomic unit (a.u.) of energy is ez/ao = 27.2114eV and the atomic unit of length

is a0 = 0.5291621.

The total hamiltonian for the system is the sum of the above electronic hamil-
tonian 7:1 and the nuclear repulsion term, VNN = i2.) Zd¢a(ZaZg/Tag), which

gives a constant contribution to the total energy.

43

We seek the solution of Hit) = Ed), via the variational method. The most
widely used form of the trial wave function of a many electron system is a Slater

determinant.

A Slater determinant can be written with antisymmetrizer operator A on the

product of spin orbitals,

t» = Al¢1(1) ...... ¢N(N)). (4.4)

Here the (ix-(7‘3) E ¢,(j)’s are one—electron functions, called spin orbitals which is

a product of a space and a spin function.

For a simple illustration, consider a closed shell of an N electron system, where
any single spatial orbital, 45,-, is occupied by an up (a) and a down (3) electron. (
See the reference [mcwe89] for the open shell case. ) The Slater determinant for

a closed shell becomes

tb = Al¢1a(1)¢w(2l ------ ¢N/2a(N - 1)¢N/23(N))- (4.5)

The electronic energy is

WWII/2) (4.6)

E =
N/2 N/2
: 22f;+:(2Jg1—I\IU) (47)
i i<j
where
Jij a (¢.(1)¢.(2)I;-:;I¢.(1)¢.(2)> (4.8)
— 1
Kij 5 <¢i(1)¢j(2llal¢i(2l¢j(1l) (4-9)

The minimization of E with respect to the functions (2'),- leads to the Hartree-

Fock (HF) equations,

f(1)¢.-(1)= e.¢.~(1) i=1,2,...N/2 (4.10)
44

where .7: is the Fock operator defined as

A A N/2 A A
HI) = f(1)+Z(2J.-(1) — 113(1)). (4.11)
i=1

where the Coulomb operator J j and the exchange operator A} are defined by

1.0) = /d..¢.(2)9(1.2)e.(2) (4.12)
113(1) = / d-v.¢.(2)i(1.2)15he.(2). (4.13)

Here 1312 is the permutation operator that exchanges the pair of electrons, 1 and

2.

In general, an orbital, (1),, called a molecular orbital (MO) for a molecule, is

written as a linear combination of one-electron atomic basis function X).

¢i = ZCuiXu- (4.14)

The basis functions )0, could be Gaussian type or Slater type centered on the
atomic positions in a molecule. Substitution of the expansion of Eq. (4.14) into

the HF equation, Eq. (4.10), gives
2 0141'qu = e. 2 at”... (4.15.)
l‘ H

Multiplying Eq. (4.15) by x: and integrating over the electronic coordinate gives
2 Cui(Fuu — eiAuu) = 09 (4.16)
B

where F,“ = (xylj'lxu) is the Fock matrix and A”, = (xylxu) is the overlap
matrix. Eq. (4.16) is called the Hartree-Fock-Roothan equation. Requiring a
non-trivial solution leads to a secular equation det(F,,,, — 5,A.,,,) = 0. Eq. (4.16)
is usually solved by an iterative process; the F“, integrals depend on the orbitals

(15,- which depend on the unknown coefficients C M’s.

45

The eigenvalues of the HF equation, en’s. are called orbital energies or one-
electron energies of the corresponding MO ass. The total electronic energy of the

system is given as

N/2 1 N/2
E = 2 Z 5; - 5 :(2Jij — 1"”). (4.17)
1 1.]

The total HF energy is given as
EHF = E + VNN. (4.18)

where VNN is the nuclear repulsion.

In a real implementation of the SCF procedure, the computer program looks
for the Cufs to give the minimum Emr, for fixed nuclear positions. Then one
often optimizes this total energy to find the best nuclear positions ( although we

will not be concerned with the latter ).

When the molecular orbitals, (A, are treated as spin-independent such as om, 2
45,3, as we did above, this procedure is called the restricted Hartree-Fock (RHF).
On other hand, if we allow 45:30 and (fir-,3 to differ, it is called the unrestricted

Hartree-Fock (UHF) procedure.

4.2.2 Beyond the Hartree-Fock approximation

The Hartree-Fock energy of Eq. (4.18) will be lowered as the basis set is improved,
approaching a limiting value as the basis set approaches a mathematical complete-
ness. This limiting energy value is called the Hartree-Fock energy. This HF energy
is however not as low as the exact energy of the system. This is because the Fock
operator 1" in Eq. (4.10) treats each electron moving in the average potential field

due to the other electrons in the system. Since the electrons repel each other, in

46

reality the movement of one electron affects the movement of the other remaining
electrons, i.e. their motions are correlated. The HF energy is higher than the
true energy because the HF wavefunction, which is a single Slater determinant,
is formally incapable of describing this correlated motion. The energy difference

between the HF energy and the exact energy of a system is referred to as the

correlation energy.

In order to overcome the restriction of a single determinant, we can introduce
some mathematical flexibility by allowing if) to contain many determinants. This
leads to one of the techniques, called configuration interaction (CI). The basic
idea of CI is simply to take the wavefunction if) as a linear combination of many

Slater determinants:

NCSF

11!: Z CIII). (4.19)

[=1

where II), called a configuration state function (CSF), is a linear combination of

determinants,
N] '
II) = 210,-»... (4.20)
i=1

The IDj)’s are single determinants with different occupation schemes. In order to
form IDj), one starts from the SCF occupied and virtual (unoccupied) MO’s. The
CSF, II), is formed from |D,-)’s to be an eigenstate of the spin operator 8'2 and
S, and classified as singly excited, doubly excited and triply excited according to
whether 1,2,3 electrons are excited form occupied to unoccupied (virtual) orbitals.
The most common type of CI calculation includes the singly and doubly excited
CSF, usually designated as CISD. After constructing CSF’s from SCF MO’s, we
minimize E = (wl’iilw) with the new 21; of Eq. (4.19) with respect to CI the

coefficients, C 1.

47

In order to increase the efficiency of a Cl calculation (fast convergence and
less computing effort ), one can construct CSF’s from non-SCF MO’s, in contrast
to the CI which starts from SCF MO’s. The multi-configuration SCF ( MCSCF
) [shep88] is one way to do that and we have used MCSCF to investigate corre-
lation effects in our cluster calculation. In an MCSCF calculation, the CSF’s are
constructed in the same way as in CI, but in the MCSCF, the configuration state
function (CSF) is allowed to vary in addition to the CI coefficients, 01’s. The CSF
can be optimized by varying the MO coefficients, Cm, of MO, p.45 in Eq. (4.14).

( It is noted that the determinant. IDJ), is constructed from these MO’s. )

For a simple 2-electron illustration of showing the difference between CI and

MCSCF, one can consider 2)) as
¢=C1|1)+Cgl2) (4.21)
where

)1) = lDll=Al¢1¢2)
l?) = ID2)=AI¢3¢.). (4.22)

(91 and (b; are occupied SCF MO’s and 453 and (94 are unoccupied SCF MO’s, which
form a doubly excited CI calculation. Each o,- is expanded in basis functions,
45,- = 2:” Cugxu. In the CI calculation one varies only C1 and C2, while in MCSCF
one varies the CI coefficients, 01 and C2, and the molecular orbital coefficients,

Cpg’s, simultaneously to optimize the wavefunction 212.

4.2.3 Natural orbitals

We will now introduce the idea of natural orbitals (NO). These turn out to be

convenient for calculating physical quantities such as charge density and spin

48

 

d1

density in terms of the one-particle density matrix in the basis of NOS.

From a MCSCF ( or Cl ) wavefunction. 111(1. 2. ....N). the one particle density

matrix is defined as

)=N/1p(1, 2,.....’V) )ti»(1.2......V)dr(2....N), (4.23)

which satisfies

/ 7(1|1)dr(1) = N. (1.21)

The electron density p(i:'1) is given as
pa.) = 1011) (4.25)

When il)(1,2, ....N) is expressed in terms of MO’s, Eq. (4.23) is written as

(1’|1)=ZZ¢:(1’A.,. (4.26)

where Ag,- is defined in Eq. (B5) of Appendix B and o.- is a spin-orbital. By a
unitary transformation, the matrix. A;,- can be diagonalized and (1’ [1 can be

written in terms of diagonal matrix elements, 12,-. and new orbitals 0.:

1l1=Z¢'1((4.27)

The spin orbitals, dg’s are called the natural orbitals (NO) and n, is the occupation

number of NO 43,-.

The expectation value of the spin operator 8'z = Z,” S,(i)6(r'— ii), where 82(1)

is spin state ( % or —% ) of i-th electron, can be written, using the one-particle

density matrix,

49

Substituting 7(1’Il) from Eq. (4.27) and Eq. (4.28) gives.

(5.) (11:1 $.(1) Z n.¢3;(1').5.-(1)dr(1), (4.29)

= /Z%0gniléi(l)l2d7'(l), (430)

where 0.- gives the spin state (+1 or —1) for up or down spin and n.- is the

occupation number of the i-th spin-orbital.

From Eq. (4.30), we define a spin density. s(f"), in terms of NO’s as
1 ~ ..
8(7?) = Z -2-o,-n,~|d>,-(r)|2. . (4.31)

As seen from Eq. (4.31), a physical quantity such as the local spin density can
be represented by a simple form using NO’s. In HF-SCF calculations, MO’s are
the same as NO’s and the n; are either 0 or 1. But in MCSCF or Cl calculations,
they are not ( and the ng’s are in general a fractional number between 0 and 1 ).
Knowledge of the natural orbitals and occupation numbers allows us to calculate
the spin density and thus the form factor. In our MCSCF calculation using the
COLUMBUS code[colu88], the NOS are not available for up and down ( a and
fl ) electrons separately, directly from the code. Therefore we have developed
appropriate programs to convert the MOS to NO’s by constructing the matrix

1451' for a and fl electrons separately. The details of developing the programs are

described in Appendix B.

4.3 Theoretical form factor

The spin density of a molecule or a cluster can be rewritten from Eq. (4.31),

OCC

3(7?) = 52% Z njdjl¢j(77)l21 (4-32)

t,-

where (13,03) is now the j-th natural orbital (NO) obtained from the HF or MCSCF
calculations; 72 = n, — 77.1 where n; (m) is sum of the occupation numbers of
up(down) spin natural orbitals, so that n = 1 for a doublet spin state and n = 2
for a triplet state. When mean field theory (MFT) for the antiferromagnetic
crystal is assumed, S. is 1/2 for a doublet cluster state and 1 for a triplet cluster

state.

We made an, ansatz that the total spin density in the antiferromagnetic (AF)
ordered crystal is given by the sum of contributions associated with each magnetic

ion ( or each of our clusters )

S(=F) 2694's -r'i), (4.33)
where n is a lattice vector associated with the chemical unit cell, and cf... is a

particular AF wave vector: 6'5”“ is +1(—1) at up(down)-spin sites T7.

The experimentally measurable quantity from the neutron Bragg scattering is
the Fourier transform of the magnetic moment density per unit cell, ngS(F)/N,

(N is the number of unit' cells) ;
1 .--
mu) = W‘s-1V /( s (me-WW

= 9113— NZ?“ /s(r— -..).—11111;

1 = 9,13— NE e-“v q""/s(1")e“f'FdF. (4.34)
The sum 2,, (“5'“)‘5 gives the Bragg peaks which are localized at the general

AF wave vectors 5);. The variation of the intensity of the Bragg peaks is controlled
by
man.) = m /s(oe-‘°‘~"dr

51

where f (cf) is the form factor defined from Eq. (4.35) and given by
1 iq'd’ ..
f(<i‘) = 3- ] .(r). 4r. (4.36)

In our cluster calculation, each natural orbital (NO) for a cluster is a linear

combination of basis functions such as

Mr”) = fo’h’) + 2 2x170"). (4.37)
k j l

where x24 is a basis function centered at the metal and xf" is a basis function
centered at the j-th ligand. The subscripts, k and l label the individual basis
functions for the metal and the ligands respectively ( for simplicity, we dropped
the index j associated with the NO’s in Eq. (4.36) ). The probability density
associated with the NO, Iqi(r”)|2 becomes

¢S(=r‘)|2 Xx). x): ”HZZXL- xf“ +2 2 xf‘xf"

k. k’ i i,l i’,l’¢i.l
+ Z: x) 'xf“ (4.33)
i l

Again for simplicity, the basis function x is assumed to include its NO coefficient.
When the i-th ligand is shared by two nearby magnetic metal ions such as Cu2+’s
with antiparallel spins in the Cqu plane. the contributions to f ((1') from the last
term of Eq. (4. 38) are cancelled by each other at q: (74 . This leads to the

well—known covalent reduction in the form factor as discussed in Chap.3.

When we take account of the spin fluctuations beyond the mean field theory
(MFT), S, in Eq. (4.35) becomes the average value of the spin in the Heisenberg
model, < S, >H¢g,, instead of the MFT value ( 1 for Ni and 1 / 2 for Cu ) [kap192]

(see Chap. 2). Thus we now have the magnetic moment density, m(r'1’A), in terms

52

of < 5, >3... which includes the quantum spin fluctuation effect, and flat.) which

differs from the mean field value by the covalence discussed in Chap. 2:

"10314) = 9/13 < 3. > f(<f.1)- (4-39)

4.4 Cluster and environment

4.4.1 Cluster

In all our cluster calculations, we have taken the basic cluster to consist of one
central metal and 6 or 5 surrounding ligand ions, (MXGW) cluster, M = Ni or
Cu and X = F or O. The geometry of the cluster, (MX6(5)), was taken from the
structure of each compound, thus it has an octahedral symmetry for KNIF3 and
NiO, tetragonal symmetry for LagNiO4 , LazCuO4 , SrgCuOgClg , and still lower

symmetry for YBagCugOs .

In order to construct molecular orbitals for a cluster. Wachters’ [wach70] basis
set, denoted as (1359p5d), was used for Ni and Cu. For F, O. and Cl, we used
Huzinaga [huzi7l] basis sets with additional diffuse p orbital, which denoted as
(956p). These basis sets, (13s9p5d) and (956p), are contracted to [7s3p2d] and
[353p] respectively ( see Ref.[levi91] for basis set denotation and contraction ).
The basis set (l4sllp6d) for Ni with additional diffuse functions suggested by
P.J. Hay [hay77] is also used to see how sensitive the calculations are to (adding
the diffuse basis functions for the metal ion. These two basis sets, (14311 p6d) and
(l3s9p5d) give the same formfactor and charge density (difference is less than
1.5 %), even though Mulliken charge values are different. ( See Appendix C for
further discussion of this point. ) Thus we present the results of the form factor
with basis set (1389p5d) for Ni and Cu.

53

4.4.2 Point charge model

In order to generate the Madelung potential around a cluster, the point charge
model was employed. We used Evjen’s method [evje67] to carry this out. Formal
ionic charges (for example in KNiF3 , +2, —1 and +1 point charges for Ni“, F“
and K+ respectively) were assigned to the atomic positions and fractional charge
values on the boundary were taken such as to make the whole system charge
neutral. For fractional charges, first we set up a cube as a Madelung cage, which
contains point charges at the atomic positions. Then if we count each ion on
a surface of the cube as being half in the cube, each one on an edge as being
a quarter inside, and each one at a corner as being one-eighth inside, the total

charge within the cube is zero.

This Evjen’s method was reported to work well in the electronic structure

calculation of perovskite and fcc structure materials like KNiFg and NiO [sou393].

4.4.3 Effective core potentiaI(ECP)

For a more realistic treatment of the extra cluster environment. we replaced the
nearby point charges by effective core potential (ECP), which enables us to incor-

porate Pauli repulsion between the electrons in the cluster and the environment.

ECP was originally developed to reduce the computational effort in generating
the valence electron wave function in an atom. by doing calculations only for those
valence electrons, instead of doing all electron calculations. The core-valence
interaction is represented by an effective core potential acting on the valence
electrons. In our calculation, we used ECP to generate a potential seen by the

electrons of a cluster arising from the nearby extra-cluster ions. Here we review

how ECP is generated as a product of Gaussian function and polynomials following

Ref. [wadt85].

The generation of effective core potentials (ECP’s) begins with numerical
Hartree-Fock orbitals. From HF orbitals. nodeless pseudo—orbitals are generated.
Then a numerical potential is rigorously obtained from the pseudo-orbitals and
then fitted to an analytic form. The underlying procedure consists of three major

steps as discussed below.

(1) Set up the pseudo-orbitals. (51's, which are smooth and nodeless, from HF
orbitals (b). The valence pseudo-orbital, (i3)(r), is chosen to behave as closely as
possible to the original valence HF orbital, (Mr), in outer region of the atom, i.e.

the valence region of the atom,

410‘) = 9M") 1‘ 2 re.
= rbf(r) 0 _<_ r < rc, (4.40)

where rc is chosen to be close to the outermost maximum of o). and f (r) is a
polynomial and b is an integer constant depending on the angular momentum l of

the orbital.

(2) Determine the angular dependence potential U) requiring <23) to be a solution
of the Schrodinger equation in the field of U) plus the Coulomb (j) and exchange

(If) operators, I7“), arising from only the valence pseudo-orbitals

[_1_d_2_+l(l+1) Z

 

2dr? 21.2 — _7‘_ + U((T') + Vval‘ ¢l(r) '3 €1¢1(T), (4.41)

Veal = Za,j(qi,-) - bif{($i)- (442)
From Eq. (4.41), U) can be determined in terms of <13) and (52’,
I(I+ 1) z 42’ 9...).
27‘2 + 7' + 241 97>: .
55

U)(r) = e) - (4.43)

 

(3) From the numerical potentials U)(r) in Eq. (4.43). the product of r2 and

U)(r) is fitted to the analytic form as

der"*exp[-Ckr2], (4.44)
k

where n). =0, 1, or 2. The parameters. (1), and C). are optimized using least squares
procedure. In general, those parameters to be used to generate ECP’s are listed

in the literature.

The total potential is represented as
L-l ‘ ‘
U(r) = mm + ZlUAr) — Ur(r)]P1. (4.45)
(:0

where P) is core projection operator as P) = II >< l I, and L is one greater than

the highest angular momentum quantum number of any core orbital.

In Fig. 4.1, the s, p,d and f effective core potentials of the La3+ ion are plot-
ted as examples of the numerical ECP’s obtained by the above procedure and
listed in Ref. [wadt85]. In the figure, the potential U)(r) — ZU/r has been plotted,
where Z” = (Z — Zcm). 'Thus 2,, = 3 for La3+ ECP. The plotted ECP’s are com-
pared with the point charge potential, -3 / r, to show the difference between them.
The electrons of the cluster experience this difference when the point charges are

replaced by ECP’s in our calculatiOns.

As seen in Fig. 4.1, the valence electrons ( or the electrons in the cluster in our
calculation ) with s or p angular momentum ( with respect to the center of the
La3+ ion ) experience strong repulsion, while the valence electrons with d and f
angular momentum experience attraction at small r. ( r is the distance from La“
ion center. ) At large r, all the potentials behave as the point charge potential,
—3/r.

56

‘ ll,l‘.‘ll"‘l‘.l.

 

 

 

 

 

 

: ' T l r U I I I I I I U I I I I I I T T I I:
4—1 _.
: I Urf')‘3/' s
n ' .
.. I -. I
s 2-1 —)
1" o _ cl
s _ : . U.(r) 3/r -(
3 : ' :
_ I '° ,.
0 ' ..."Oo..."
g l U4(')-3/r
no ' .fl”
’2 1 ,r’ "J
)- ‘ ’ .‘o d
'- \‘ I .3." U!(') " 3/' ‘1
..4 ~— -3/r.-'. -:
n V .'
"' . L L1 L1 L1 1 1 I L L1 [1
0 1 3 3 4 6

:- (a.u.)

Figure 4.1: The .9, [1,41 and f effective core potentials for La3+ are plotted in the
form, U)(r) - 3/r. These potentials behave as the point charge potential, —3/ r in
the large r region.

57

Chapter 5

Neutron scattering form factor
of the nickel compounds

5.1 Introduction

Since we decided to carry out a more realistic calculation of the neutron scattering
form factor using ab initio methods, we first studied the weakly covalent anti-
ferromagnets KNiF3 and MD to test our cluster model calculations. and then we
applied the same method to LanlO4. Several ab initio cluster calculations have
been done in KNIF3 [elli68, mosk70, soul71, wach72] and NiO [bagu77, jans88].
However, most of these works were concerned with the excitation energies rather
than the ground state wave function, which is needed to obtain the neutron form
factor. The only cluster calculation of the form factor was carried out for KN iF3
[elli68] more than 20 years ago, and was considered by the authors to be too crude
to even compare with the experiment. Other attempts, which had been made to
get the theoretical'form factor using the free ion Ni“, wave function didn’t give
good agreement [shar76] with experiment. This paucity of work on the theoretical
form factor motivated us to carry out cluster calculations using the techniques of

ab initio quantum chemistry which have developed rapidly during the past two

58

decades [kap194]. We followed a standard procedure, namely the basic cluster was
chosen to contain one Ni2+ ion and its 6 nearest neighbor ligands ( F" ion or 02"
ion respectively), and the rest of the lattice was taken into account by employing
the point charge model explained in Sec. [4.4.2]. Corrections to the point charge
model in the form of limited Pauli repulsion were also considered by taking ECP

of the nearest extra-cluster cations.

In this chapter, I discuss the results of the cluster calculations for Ix'NiF3,
NiO and LagNiO4 carried out with both RHF and UHF methods [chan94]. Then
I compare our theoretical results with experiment in these nickel compounds.
Especially for KNiF3 , where absolute experimental values of Bragg scattering
are available, we could estimate the quantum spin fluctuations by fitting the

theoretical values to the experimental values.

5.2 KNiFg

KNiF3 is an antiferromagnetic insulator and is considered to be an ideal compound
to test theory vis-a-vis the experiments from several points of view[huch70]. It
is well-known that pure stoichiometric samples of this cubic perovskite structure
material can be prepared even at low temperature. The magnetic structure of
KNIF3 is known as simple cubic G-type[scat61]. In the G-type structure, a par-
ticular magnetic ion is coupled antiferromagnetically to its six nearest magnetic

neighbors. (The magnetic ions form a simple cubic structure.)

In neutron scattering experiments, the nuclear diffraction intensities are gov-
erned only by the known nuclear scattering lengths. Therefore, by comparison,

these nuclear peaks may be used to obtain the magnetic cross section on an ab-

.59

solute basis [huch70]. The magnetic form factor measurement, in other words
the measurement of magnetic peak intensities for more than one Bragg peak, was
carried out by two groups [huch70, scat61]. The later experiment by Hutchings
and Guggenheim [huch70] gives absolute values of the intensities of the magnetic
peaks. Therefore we have chosen Hutchings and Guggenheim’s experiment to

compare with our theoretical calculations.

5.2.1 Experimental form factor

We have determined the form factor from the measured intensities of neutron
powder diffraction in KNIF3 in Ref. [huch70]. Hutchings and Guggenheim mea-
sured the Bragg peak intensities at 4.2 K. The lattice parameter of the chemi-
cal(nuclear) unit cell shown in Fig. 5.1 is ac = 4.0024. But the lattice parameter
of the antiferromagnetic unit cell is doubled from that of the nuclear unit cell and
is 200 = 8.0021. The magnetic peaks are indexed according to the magnetic unit

cell.

Experimentally, one can determine 9 < S. > f (5M) from the ratio of nu-

U

clear and magnetic Bragg peak intensities . This quantity is related to several

experimentally measured quantities by the equation ( see Ref. [huch70] )

‘ 2
IMzNFfi 223,, 2mc2
IN ‘03, zMFfilel e27

 

{9 < 5: > f(thll2 =

exp{ —2B(sinO~/A)2} sinOMsin20M
exp{—2B(sinOM/A)7} sinONsin2ON

 

(5.1)

Here IM and IN are magnetic and nuclear peak intensities and the subscripts,
M and N stand for magnetic and nuclear Bragg peaks respectively. B is an

average temperature parameter, and v... and 22,, are the volumes of magnetic

60

 

 

 

 

 

 

 

 

Figure 5.1: Crystal structure of KNiF3 . The small (solid) circle is Ni“ ion, and
the middle-sized (empty) circles are K+ ions, and the big (solid ) circles are 02-
ions. The (NiF5)"" cluster consists of the solid circles.

61

Table 5.1: Experimental values of < Sz > f(q'iu) in KNiFg

 

 

magnetic peak index < 52 > f((j'M)

 

(111) 0.783
(311) 0.672
(331) 0.552

 

 

 

 

.= rt

 

and nuclear unit cells respectively ( vm = 81),, ). ‘fM are the antiferromagnetic
Bragg-peak wave vectors, called (i4 earlier in this thesis. The other parameters
of Eq. (5.1) are structure parameters depending on the peak index. Hutchings
and Guggenheim [huch70] measured 3 magnetic Bragg peaks , (111),",(311)M
and (331) M and they also measured the nuclear (200))v peak which is well sep-
arated from nearby magnetic peaks. Thus we chose 1200 as a reference nuclear
peak to calculate the magnetic form factor f(tj'M) in Eq. (5.1). We calculated
the quantities, [g < S. > f(tj’M)/F'goo]2 for the 3 magnetic Bragg peaks using
F111 = F311 = F331 = 8,, N1211 = N3“ = N3231 = g, 2111 = 8,23” = 24, 2331 = 24
and 2200 = 6. Here (iffy = 0.2695 x 10“A and 9’s can be obtained from Fig.
1 in Ref. [huch70] as 9111 = 7.0°,0311 = l3.5°,0331 = 17.75° and (9200 = 825°.
The quantity, [9 < S, > f (6M) / F200] now involves no unknowns and so has only
random experimental error. F200 was determined from scattering lengths as
F200 = bN, + (bp - bx) and was given as F200 = 1.218(21:0.020)Cm‘12 in the
same reference. Taking g = 2.29(:t0.02) from Ref. [huch70], we obtained the

values of < 52 > f(tj‘M) which are listed in Table 5.1.

The quantity, < Sz > f (6114) is governed by quantum spin fluctuations(QSF)
and covalence(COV) simultaneously as we discussed in Chap. 2. But Hutchings

and Guggenheim’s interpretation of the quantity, < S: > f((j’M), is different
62

from ours. They took f(tiu) at the first Bragg peak (111)," from Alperin’s NiO
form factor and then interpreted < 52 > to include both QSF and COV effects
assuming f (6M) is normalized to 1 when extrapolated to [JMI = 0. According to
Hubbard-Marshall theory [hubb65]. the antiferromagnetic form factor shouldn’t
be normalized to 1 when extrapolating to WM) = 0 because of COV as discussed
in Chap.3. If we knew f(q'iu) which includes COV, then < 5. > should be
interpreted as < 5, >33), as discussed in Chap. 2. We have therefore calculated
the form factor f (q"M) and scaled our calculated values to give the best fit to the
experimental quantity < S. > f((fM) which then gives us < 5; >36), directly.

This is compared with the spin wave theory later in Sec. 5.2.4.

5.2.2 (NiF6)4" cluster

The (NiF6)4‘ cluster in perovskite KNiF3 has octahedral symmetry as shown
in Fig. 5.1 . The distance between Ni2+ and F" was taken as 2.00/01. RHF and
UHF Self-Consistent-Field (SCF) calculations on this cluster, were performed with
the COLUMBUS code[colu88] and the Gaussian 92 code [g92] respectively using
contracted Gaussian basis sets. All the electrons of the cluster. 86 electrons, are

explicitly included in these ab initio calculations.

Huzinaga [huzi71] basis sets (936p) with additional diffuse p function are used
for F. The basis set for Ni is Wachters’ [wach70] basis (13s9p5d). The basis set
(14sllp6d) for Ni with additional diffuse functions suggested by Hay [hay77] is
also included to see how sensitive the calculations are to adding the diffuse basis
functions. These two basis sets, (14sllp6d) and (1389p5d) give the same form
factor and charge density (difference is less than 1.5 90), so we present the figures
of the form factor obtained with the basis set (13s9p5d) of Ni ( see Appendix C

63

Table 5.2: Madelung potential value at the origin of the cluster in KNiF3

 

 

number of point charges“ potential (eV)

 

82 -21.25
192 -2l.16
482 -21.05
784 -20.98
1434 -20.96

 

 

 

==

 

“This number of charge does not include the cluster itself.

for further discussion of this point ).

5.2.3 Environment

For the rest of the system outside the cluster, the point charge model is employed
according to Evjen’s method [evje67] discussed in Sec.[4.4.2] We took 482 point
charges to obtain the Madelung potential for NiFg" in KNIFg, after establishing
a reasonable convergence in the value of the potential at the center of the cluster

( variation within less than 0.2 ‘70) shown in Table 5.2.

It should be noted that 482 point charges correspond to 64 chemical unit cells.
The potential value generated by 482 point charges in KNiF3 by Evjen’s procedure
is known to give an almost constant difference from the exact Madelung sum even
rather far from the origin of the cluster for the perovskite KN 1F 3 [sous93]. The
constant difference of the Madelung potential does not affect, of course, the motion
of an electron in the potential. We found that the calculated form factor hardly
changed by increasing number of point charges beyond the 64 unit cells. The

crystal field splitting of d electron of Ni2+ are discussed in Appendix [D] also

64

hardly changed either.

For a more realistic treatment of the environment effect in KN IF3, the point
charges which originally represented the 8 nearest neighbor K+’s were replaced
by Effective Core Potentials (ECP)[wadt85] in Sec.[4.4.3]. This enabled us to
incorporate Pauli repulsion between electrons in the F"s of the cluster and those
in neighboring K+’s. The effect of this replacement on the form factor of KNIF3

was found to be negligible so we do not present the calculation including K+ EC P.
5.2.4 Comparison with experiment

The theoretical AF form factor values from RHF and UHF are compared with
the experimental values of Hutchings and Guggenheim [huch70] in Fig. 5.2-(a),(b).

The experimental values of the product, < S’z > f (q'ju) were taken from Table 5.1.

The calculated form factor values are multiplied by the factor < S, > to fit the
experimental data. The UHF results in Fig 5.2-(b) differ slightly from the RHF
results in Fig 5.2-(a), but this small difference helped to obtain a near perfect
agreement between the UHF results and the experiment. The best fit to the
experiment in Fig. 5.2-(b) gives < 5', >= 0.90 which can be directly compared
with the result of spin wave theory for the simple cubic lattice < S, >,p.-nwa,,e=
0.92 [ande52]. The covalent reduction in KNiF3 was found to be 0.95 in UHF
calculations. ( See Eq. (2.68) for the definition of the covalence reduction factor.)
We conclude that the theoretical results for the magnetic moment density m(r}'A)
in terms of covalence and spin fluctuation effect (Eq. (4.39)), agree very well with

the experimental data in KNiF3.

65

 

 

 

 

 

 

1,0 ....,....,....].-..,.... ...-.l....,.e..1....lt..e
L {1- 4
l ’3 1. .
1- “ «(b J
. 3 c: .. .
AM- 4: a; -- :1: -
m )- v d:- «t
)- A <1- 1
1 )- é 8 <1- i 4
v P 8 j). s
0.6b -- -
(- a s- * 4
:1 I II I
tal/0.4”- ‘f- -
A . .1 .
m" .DRHF ). XUHF .
V 1- . 4L . .1
0.2 - fr experlment f experlment
p 1r '4
: '(a) 1: (b)
00 ...il...il....l...il.... ...il.i..l....l....l..
'o 1 2 a 4 0 1 2 a 4 -5

4nsin0/A (71“)

Figure 5.2: The calculated (a)RHF and (b)UHF form factor are compared with
Hutchings and Guggenheim’ experiment [huch70] in KNiF3 .

66

5.3 NiO

After the success of ab initio cluster methods in calculating the neutron form factor
in KNiF3 , we decided to apply the same procedure to NiO before proceeding to
LagNi04 . NiO had been studied for a long time both theoretically and experimen-
tally for various physical properties. But no calculation of the magnetic form fac-
tor was available with ab initio or any other sophisticated theoretical methods even
though experiments had been done by Alperin more than 30 years ago[alpe61].
The form factor of NiO has been measured at many Bragg peaks [alpe61] while
in KNiF3 , experiments were available for only 3 Bragg peaks[huch70]. In fact
the Bragg peaks over a broad range of |q"| can give detailed information about the
spin distribution in |7"‘l space. Thus NiO is an excellent case to test our cluster

method.

Just like KNIF3 , NiO is an antiferromagnetic(AF) insulator with TN = 525K.

It has the rock-salt structure as shown in Fig. 5.3 with a0 = 4.1674.

The N i“ ions along the line Ni-O-Ni have antiparallel spins while the nearest
neighboring Ni2+ ion spins are parallel. The spins of magnetic ions are aligned
in the (111) plane forming a ferromagnetic sheet and each (111) ferromagnetic
sheet is antiparallel to the adjacent (111) sheets[mart67]. The AF unit cell has
the lattice parameter 200 = 8.32/4 with a fcc structure and its volume is 8 times
the volume of the chemical unit cell. The Bragg peaks in N iO are indexed by that

magnetic unit cell.

67

 

 

 

 

 

 

 

 

 

 

 

Figure 5.3: Crystal structure of N10. The small circles are Ni2+ ions and the big
circles are 0" ions. The (NiO¢)‘°' cluster consists of the solid circles.

68

Table 5.3: Madelung potential value at the origin of the cluster in NiO

 

 

number of point charges“ potential (eV)

 

118 -24.26
326 -24. 19
722 -24.20
1324 -24.20

 

 

 

 

 

 

“This number of charge does not include the cluster itself.

5.3.1 Cluster

The (Ni06)l°‘ cluster was chosen: it has octahedral symmetry as shown in
Fig. 5.3. The distance between Ni and O is 2.08.31. All the 86 electrons of the
(Ni06)1°' cluster were taken into account in the RHF and UHF calculations.
COLUMBUS [colu88] and Gaussian 92[g92] codes were used for the RHF and
the UHF respectively. Wachters’ basis set (13s9p5d) for Ni[wach70] was used as
in KNiF3 and Huzinaga basis sets (956p) for O [huzi71] with an additional dif-
fuse p function were used. The Mulliken charge for N 10 was found to be very
sensitive to the choice of the basis set for Ni. But physically significant quanti-
ties, the form factor and charge densities hardly changed from basis set (13s9p5d)
to (14sllp6d). The apparent puzzle indicated by these results is discussed and

resolved in Appendix C.

5.3.2 EnvirOnment

The point charge model by Evjen’s method in Sec.(4.4.2) was taken for the rest
of the system outside the cluster. 722 point charges within 64 chemical unit cells
were taken to produce the Madelung potential in the region of the (N106)1°‘

69

cluster. The potential value at the center of the cluster is listed in Table (5.3) and
as can be seen in the table, the potential is well-converged by 722 point charges.
Evjen’s method with 722 point charges (even 336 point charges) in a fcc structure
like NiO was found to be a good approximation to the exact Madelung potential
in the electronic structure calculations [sou593]. Since the effect by adding ECP
was found to be negligible in KNIF3 , we did not incorporate ECP in the NiO

case.
5.3.3 Comparison with experiment

The theoretical AF form factor values from RHF and UHF are compared with

Alperin’s single-crystal experimental values [alpe61] in Fig. 5.4 and Fig. 5.5-
(a))(b)-

The absolute values of < 5, > f (@241) are not available from Alperin’s experi-
ment; the experimental values were scaled by 0.93 to give the best fit to our UHF
results, particularly in the small l‘l'i-tl region. The experimental Bragg scattering
data in NiO extends to a larger region of ({1}) compared to that in KNIF3. In the
Fig. 5.5-(a), we compare our UHF results with the scaled experimental values.
The UHF results agree very well with the experiment for the first three Bragg
peaks and are consistently somewhat lower than the experiment for the larger
((7.1) values. However, the bumpiness of the data is traced rather well by our theoo
retical calculations, which results from the asphericity of the spin density around
each Ni. The overall agreement between the UHF results and the experiment in

Fig. 5.5-(a) is reasonable.

An additional contribution to the form factor comes from the orbital motion

of the electrons. This contribution was found to be appreciable only for large (til

70

 

1.0

j

' (inn (iui (iaialn' 'fm') (imito'ui Micah-711' l ' '

f(qi)

 

 

 

L -(
I (as) (m) (as) (m) (11.4.1) (an) I
- 3' . um (um . (13.1.1) -
0.8 — (as) (so) (no) (114311111141)—
: 5 (an) 1
)— -(
0.6 :— 6 i '7,
0.4 :— i ‘j
.. D RHF 0 ¥ {1 d
" i o t D i D :
0 2 'I-_ experlmen CID“ ‘ i a
; D 8*‘1’5 a j
0 0 b 1 1 1 1 l 1 1 1 1 l 1 1 L 1 f 1 1 1 10' m
0 2 4 6 8 10

Amino/1t (71“)

Figure 5.4: The calculated RHF form factor is compared with Alperin’s experi-
ment [alpe61] in NiO.

71

 

 

 

     

 

 

1.0%---.I....,..-.lNWT.“
I 9
0.8: a
0.6:— as
0.4:-
; DzUHF 5%, E
02E' izExperiment 0530 11
E..(e>......... 5;
30.0:V 'fl'rrvlru
I 1
1." 0.8;- i
0.6}- *i
0.4C-X:UHF 1
: + orbital contributio X * x
0.2 E"! : Experiment. § i ¥ ;
: ml 1 1 *g
0.01111 11111111 1111 11
O 2 4 6 8 10

‘41rsin9/A (ll-1)

Figure 5.5: The calculated (a) UHF form factor, and (b) UHF form factor with
orbital contribution from Khan et. al.[khan81], are compared with Alperin’s
experiment[alpe61] in NiO.

*1
l0

in Ref. [khan81, blum61]. We took the orbital contribution for NiO from the work
of Khan et. al. who made a spherically averaged estimation of this contribution
using the ionic Ni“ wave function [khan81]. This orbital contribution is negligible
in the small q region, so we do not expect it to be important in our discussion
of the KNiF3 results, where the experimental data are available only for small
q. For larger q, the orbital contribution in NiO helps to give a better fit with
the experiment, as shown in Fig. 5.5-(b). The small discrepancy between the
calculated and the experimental values in Fig. 5.5-(b) might arise from the error
involved in the spherically averaged estimation of the orbital contribution. With
the inclusion of the orbital contribution to the form factor, we conclude that the

results in Fig. 5.5-(b) are in very good agreement with experiment.

Unfortunately, we could not determine < 5z >35, from Alperin‘s experiment
by scaling the calculated form factor as we did in KNiF3 , because the experiment
[alpe61] gave only the relative form factor values. However we found a later
experiment by Fender et. al. [fend68] who measured only one magnetic peak, but
who gave information which allowed a determination of the absolute intensities;
the ratios of intensities for the (111), (222), and (400). From the intensity ratio,
they determined ($23): which can be written in terms of the scattering lengths,

exp

(MW and 00,

(gig) = (i) 32(b:,-21:b0)' (5.2)

Here (111) and (23.2) refer to the magnetic peak and nuclear peak respectively.

 

p = (e27/2mc2)gSf where 7 is the magnetic moment of the neutron, g is the
Landé factor, f is the magnetic form factor and S can be interpreted as < S: >g,,,.

Using the values, (m, = 1.03 x lO’lzcm, (20 = 0.577 x 10"”cm and g = 2.23 from

73

the same reference and taking the magnetic form factor fm from our UHF cluster
calculation as fur = 0.871, we obtained < Sz >Hm= 0.88. We also estimated
< 5, >11“, as 0.87 from (%):IP given in the same reference. Considering the
spin wave value, < S, >Heg,= 0.92, for Type II AF fcc structure as appropriate
for Ni0[coll72], our theoretical estimate, < S, >Hm= 0.88 i 0.01 appears to be
quite reasonable. In our UHF calculation for NiO, the covalence reduction factor
in Eq. (2.68) was found to be 0.91. Considering the covalence reduction factor
in KN1F3 , 0.95, we conclude that NiO is more covalent than KN1F3 . This is
physically reasonable because the 02" wave functions are more diffuse compared

to F' wave functions.

5.4 LagNiO4

Now we apply our ab initio cluster method to calculate the form factor of La; N iO4.
The procedures of the cluster calculation are very similar to those of N iO. The
main difference between the clusters NiO and in LagNiO4 is the structure of the
cluster and the surrounding point charges. The cluster for LagNiO4 is tetragonal

while the cluster for NiO is octahedral.

Recently L32N104 has received special attention since it is isostructural with
LagCuO4 which is the parent of a high Tc superconductor. It also exhibits a struc-
ture distortion from tetragonal to orthorhombic at ~ 700K similarly to La2CuO4.
However, LagNiO4' shows another structural transition from orthorhombic to low-
temperature tetragonal structure(LTT) at ~ 70K[land89]. Also doped LagNiO4

has been found not to be a superconductor.

The crystal structure of LagNiO4 is shown in Fig. 5.6. Wang et. al. [wang92]

74

 

 

Figure 5.6: Crystal structure of L32N104 . [18.201104 has the isostructure of
LagNiO4 (Ni2+ ions are replaced by Cu2+ ions ). The contents of the (Ni06)‘°‘
cluster are connected by the dotted lines.

75

measured the form factor at 1511' where LagNiO4 is in the LTT phase. We followed

their indexing for magnetic Bragg peaks for LagNiO4 .

Several values of the Néel temperature, TN, in LagNiO4 have been reported
in the literature. TN was reported «to be 650K in Ref. [land89] and 330K in
Ref. [naka95] even for stoichiometric L32N104 , as the authors of the latter pa—
per claimed. TN is very sensitive to oxygen content in LagNiO4 : L32N104+005
shows TN = 70K in Ref. [aepp88]. Although the value of TN is controversial, the
temperature, 15K, where Wang et. al.[wang92] measured the form factor, is well

below any of the TN values.

5.4.1 Cluster

The tetragonal cluster, (Ni06)‘°‘ , was taken for LanlO4 with distances between
Ni-O as 1.95121 in the NiO; plane and 2.2121 along the apical axis. Wachters’
basis set (1339p5d) for Ni[wach70] and Huzinaga basis sets (956p) for O [huzi71]
with an additional diffuse p function were used as in NiO. 'All 86 electrons in
the (Ni06)1°' cluster were taken into account in our RHF and UHF calculations.
The COLUMBUS[colu88] and Gaussian 92[g92] codes were used for RHF and

UHF respectively.

5.4.2 Environment

We took 552 point charges to simulate the crystalline environment around the
cluster in L32N104. These 552 point charges for LagNiO4 are within 27 unit cells
which is denoted as PC333 in table 1 of Ref. [mart91]. The deviation from the
exact Madelung sum is listed in the same table. We noted that this number of

point charges for LagNiO4 does not give a constant difference from the Madelung

76

sum; however it did not seem to affect the form factor. Increasing the number of
point charges, from PC333 (27 unit cells) to PC553 (75 unit cells ) in the notation

of Martin’s paper, hardly changed the form factor.

Although the point charge model was found to be good enough to generate the
Madelung potential for KNng or N iO, we speculated that the point charge model
might not be appropriate for L32Nl04. The calculated form factor with different
ECP environments is shown in Fig. 5.7. Since the nearest neighbors of the cluster,
La3+’s, are quite close to the apical oxygens, a point charge, +36, for La3+ might
attract the electrons of the apical oxygens too strongly. The Mulliken population
analysis from our cluster calculations on LagNiO4 showed that more charges were
assigned to the apical oxygens than the planar oxygens, which is consistent with
the above picture. Bare point charges are of course, more attractive than real
ions. One way to reduce this strong attraction by the La3+ in a realistic way
is to introduce an effective core potential (ECP) for these nearby La3+ ions. As
a first step to include ECP, we replaced the eight nearest neighbor +3e point
charges by La3+ ECP’s which is denoted as ECP 1. It did change the form factor
as shown in Fig. 5.7 and Mulliken charge of the apical oxygen was reduced. As
a second step, denoted as ECP2, we replaced two more +312 point charges, right
above and below the apical oxygens by La3+ ECP in addition to the 8 La3+ ECP’s
considered above. This procedure including 10 La3+ ECP’s also changed the form
factor and reduced Mulliken charge of the apical oxygen. Then we replaced the 4
point charges of +21: with Ni2+ ECP [sabe80] for neighboring Ni2+ ions, noted as
ECP3. Changing from ECP2 to ECP3 hardly changed the form factor.

f(q1)

 

 

1,0 rm,.-.rr....,.-..,....,-... m.I.m,....,....,....,....‘
. 0 ,

. X ,
0.8 ’- B —1
F 1

p a

P 4
0.6 r- -
P ‘5 ‘

" 4

r a 1
0.4 - a -(
- 1

4

0 ECPO a

0.2?- x ECPI

 

 

 

00 AALLIAAAAIAAAAI AA AAA IAULLAAAILJLL
e

 

41rsin9/A (3.“)

Figure 5.7: The calculated UHF form factor of L32N104 with different ECP envi-
ronment were compared. In ECPO, only point charge distribution is considered,
i.e. no ECP incorporated. In ECPI, ECP2, and ECP3, (8 La“ ), ( 10 La3+ ),
and ( 10 La” + 4 Cu” ) point charges are replaced by the appropriate ECP’s
respectively.

5.4.3 Comparison with experiment

The theoretical AF form factor values with ECPI environment from RHF and
UHF are compared with the experimental values of Wang et. al.‘s [wang92] in
Fig. 5.8. Both RHF and UHF results in Fig. 5.8 disagree seriously with the
experiment. Especially, the experimentally observed plateau at small q values is

not reproduced in either calculations or other ECP environments shown in Fig. 5.7.

The effect of the spin fluctuations is only to scale the calculated values by a
constant factor; we clearly can not reproduce the shape of the form factor with

this type of scaling.

The plateau at small q in the measured form factor was seen not only in
LagNiO4 but also in LaQCuO4[fre188]. If we assume that this plateau characterizes
the covalence effect on the spin density through the Ni-ligand cross terms, it
appears that our present cluster model has failed to describe this covalence effect

in LagNiO4, in contrast to the success in KNiF3 and NiO.

5.5 Summary

The remarkably good agreement between our UHF results and the experiment
in KNiF3 and NiO indicates that the UHF cluster method, with a simple point
charge model, well describes the form factor for these rather highly ionic materials.
For KNlFa, where the absolute experimental values are available, we found that
the experimental data support our previous theoretical studies [kap192], namely
the magnetic moment density m(c]',1) is affected by both the covalence and the
quantum spin fluctuations. Furthermore, the reduction due to the spin fluctu-

ations agrees well with the spin wave theory. For NiO, we conclude that the

79

 

1.0 Illllllllllllf TUITFITIT llrl

 

 

0.6

X
0.6 1::
f

f}; ff

“(1.1)

EX
if

*1 exp eriment
D RHF
>< UHF

0.2

llllrlllll11111ll11lllll

""J l l
1111 11,1111g11LLv1111

 

 

 

0 1 2 3 o 4 5 6
4nsin6 />1 (A71)

00 llllllllllllLLlLlllllLlLllllL
0

Figure 5.8: The calculated RHF and UHF form factor are compared with Wang
et. al.’s experiment[wang92]. These calculation have been done with ECPI envi-
ronment in the text.

80

calculated values of m(cf,1) which include an approximate evaluation of the rather
small orbital contribution are in excellent agreement with the experiment. For
further improvement in the theoretical results, we need to use a more accurate
calculation of the orbital contribution (in the larger Irfl region). We also estimated
< 5, >11“, for NiO using our calculated form factor and found it to be close to

the spin wave value.

The covalent reduction of the ordered moment as defined in [kap192, maha93,
hubb65] is found to be 0.95 and 0.91 for KNiF3 and NiO, respectively. These val-
ues were obtained from the UHF calculations using the basis set (1359p5d) for Ni.
When we used the basis set (14sllp6d) which includes diffuse basis functions for
Ni, we found slightly different values, 0.92 and 0.88 for KNiF3 and NiO, respec-
tively. We understand that when the diffuse basis functions in Ni give appreciable
density on the ligands, a simple subtraction of the v(1"')2 terms in Eq. (4.38) and
the appendix of Ref. [maha93] leads to a different value of the ordered moment.
Thus the ordered moment defined in Refs. [kapl92, maha93, hubb65] depends on
the choice of the basis set like the Mulliken charge population discussed in Ap-
pendix C. The ordered moment is defined to be determined by propagating the
cluster spin density along AF ordering in the crystal and integrating the spin
density in the Wigner-Seitz cell in real space. But it needs enormous numerical
calculation. However the estimation of covalence by the above method, i.e. via
the subtraction of the )((1"')2 terms, can be a reasonable way if a compound is ionic
and the basis set is chosen properly. We believe the values from the basis set
(13s9p5d) are more reasonable in the view of the reasonable Mulliken charges of

that basis set.

In LagNiO4 , the result of the essentially similar cluster method completely

81

failed to capture even the essential qualitative feature of the form factor exper-
imentally. The difference between the clusters in N i0 and LagNiO4 is structure
and environment. The AF ordering is 2-dimensional in LagNiO4 while it is 3-
dimensional in NiO. Therefore the magnetic moment of the apical oxygens is
canceled by AF ordering in NiO but not in L32NlO4 . The form factor of LagNiO4
seems very sensitive to the spin density of the apical oxygen because it shows a
noticeable change by replacing nearby point charges by La” ECP. The simple
cluster with point charge plus ECP might be inadequate for LagNiO4 . We ap-
plied a similar cluster method to La2CuO4 and the difference between Ni and Cu

will be discussed in Chap. 6. The failure in LagNiO4 is also discussed further in

Chap. 6.

Chapter 6

Neutron scattering form factor
of the cuprate compounds

6.1 Introduction

In this chapter we discuss our ab initio calculations of the form factor in the
cuprate compounds, La2CuO4 , Sr2CuOgClg and YBazCu3Os . We followed the
same approach as in our earlier work on nickel compounds in discussed in Chap. 5
because of our success in KNiFa and N i0 in comparing our calculations with very
detailed and extensive experimental data. Although problems with LagNiO4 still
remain as discussed in Sec. [5.4], we believe a comparison between (Ni06)‘°‘ and
(Cu06)1°‘ clusters might give some insight into the difference between Ni and Cu

compounds and possibly the source of the trouble in the nickelate.

We have carried out both RHF and UHF calculations on the clusters (C 1106)“)-
, (CuO4Clg)8'and (Cu05)8‘ for La2CuO4 , Sr2Cu02Clg and YBagCugOe , re-
spectively. The ions outside the cluster are treated as point charges, except that
nearby external ions are replaced by effective core potentials (ECP) similar to
L32N104 . We also extended our calculation method beyond Hartree-Fock (HF)

to include correlation corrections via the multivconfiguration self—consistent field

83

method (MCSCF).

In addition to the cluster form factor, we have also calculated the Cu2+ ion form
factor using various methods, RHF, UHF, MCSCF and CI. These calculations
of the ionic form factor enabled us both to study the effect of covalence (by
comparing with cluster form factor) and to evaluate the reliability of estimating

the correlation effect using MCSCF and configuration interaction (CI) methods.

In this Chapter, I discuss the result of calculation of the Cu2+ ion with various
methods. Then I address the methods and results of our cluster calculations and
compare the results with experiment on YBagCugOe , LagCuO4 and szCLlOzClg .
I conclude the chapter comparing the result of the cuprates with the corresponding

nickelates.

6.2 Cu2+ ion

Since MCSCF results of the clusters, (Cu06)1°" and (Cu05)3", indicated that
d — d intra-atomic correlation of Cu2+ in the cluster was most important (as
discussed in the later sections), one might assume that the correlation effects
in a free Cu2+ ion are similar to those in the (C1106)1°‘ and (C1105)8‘ clusters.
Accordingly, we decided to study in detail correlation effects in the free Cu2+ with
various levels of approximation. We have carried out UHF and single- and double-
substitution CI (CISD) calculations using Gaussian 92 [g92]. Two levels of CISD
have been carried out and will be denoted as CISDI and CISD2 respectively. We
correlated only the 3d electrons in CISDl and 3s, 3p and 3d electrons in CISD2.
In addition, RHF and MCSCF calculations for the Cu2+ ion have been carried out

with COLUMBUS [colu88]. In this case, MCSCF allowed only the 3d electrons

84

 

'h‘

to be correlated similar to CISDl. When we say that we correlate 3d electrons,
that means we choose 3d-dominant occupied molecular orbitals (MO) and other
virtual (empty) MO’s to build determinants in the configuration state functions

( see Chap. 4 ).

To check the sensitivity of the form factor to the choice of basis set for the
Cu2+ ion, the basis set (14sllp6d) with additional diffuse functions suggested by
Hay [hay77] was used instead of the (13s9p5d) basis set. These two basis sets,
(14sllp6d) and (13s9p5d) contracted to [854p3d] and [753p2d] respectively, gave
almost the same ion form factor. Therefore We present only the results with the
basis set (13s9p5d) for the Cu“ ion. (See Appendix C for details about choosing

different basis sets.)
6.2.1 Ionic form factor

We have calculated the form factor from the ionic wavefunction obtained by RHF,
UHF, MCSCF, CISDl and CISD2 for the (1‘ values corresponding to the Bragg
peaks in YBagCugOe which Shamoto et. al.[sham93] measured for two fami-
lies, (1/2, 1/2, k) and (3/2, 3/2, h). This notation refers to a tetragonal structure,
(a,a,c), taking a along the line between the nearest neighbor Cu+2 and c along

the apical axis.

The RHF and UHF results are compared in Fig. 6.1-(a) and the UHF and
MCSCF results are shown in Fig. 6.1-(b). The form factors from CISDI and
CISD2 were found to be almost same as the MCSCF results, so here we present
the latter only. Total energies of the Cu2+ ion obtained with these various methods
are listed in Table 6.1. The lowering in the total energy due to correlation is about

5 eV and does not depend strongly on the environment.

85

 

1,0-...Tfn.,....I..h.--.l....,....,.
.- XX 1. m
: ¥¥X (1/2 1/2 1‘) 2' Ban
0.8- +15 -— a
C 15 I a
. x .. a
.. + J.

0.6: 3f- -~ 2!
a 1- + xx 4» B
30.4 r- (3/2 3/2 1:) ++ 'T' U

4»
.. J.
- X UHF 4- X UHF
0-2 T + RHF “.7 n MCSCF, CI

 

 

.l l

A A L I L L L

cl-i

 

 

:- db
00LLLlllLlllllllllAllLlllAlllllLlLl
.
..

(a) . (b)
41rsin0/A (ll-1)

Figure 6.1: The calculated ionic form factor with RHF, UHF and MCSCF meth-

ods.

86

Table 6.1: Total energies (eV) relative to the RHF values of various calculations
for Cu“ ion; It is noted that the RHF energy value includes the nuclear repulsion
amongst the environmental point charges as well as between the ion nucleus and
the point charges.

 

 

 

 

= : 44:;
free Cu” ion Cu2+ ion with point charges “
RHF 0.0 b 0.0 c
UHF -0.011 -0.013
MCSCF -2.414 -2.514
CISDI -3.031 -3.088
CISD2 -5.403 -5.454
L

 

 

 

 

 

 

“Point charges correspond to the atomic positions of YBagCugos
”RHF energy = -l638.01987633 (a.u.) ( l a.u.=27.2114 eV )
“RHF energy = 488685744701 (a.u.)

The differences between the RHF and UHF form factors are noticeable except
for the first few points ( at small Id] ) in Fig. 6.1-(a). When we considered the
contribution to the form factor from each atomic orbital, we found the difference
between RHF and UHF mainly came from the spin density of the paired electrons
due to core polarizationiin UHF (including the tgg electrons). At small Id], corre-
sponding to large If], the form factor picks up mainly the spin density from the
3d orbitals. Thus the contribution from the core polarization of 3s and 3p to the
form factor is negligible in the small Icfl region. This explains the result that the

difference at small If] is smaller than at large ltfl.

We found that the MCSCF form factor hardly changed from the UHF results
as seen in Fig. 6.1-(b). However, as shown in the Table 6.1, although the total
energy value changed by only about 0.01 eV from RHF to UHF, it changed by
more than 2 eV due to the correlation effect involved in MCSCF and CISD. That

is, the core polarization associated with UHF affects the form factor (i.e. the

87

spin density) appreciably but not the energies, while correlation effects included

in MCSCF, CISDl or CISD2 hardly affect the form factor, at least in Cu2+ ion.

We have calculated the form factor for the Cu“ ion with point charge distri-
butions of YBazCugOe using RHF, UHF, MCSCF, CISDl and CISD2. The form
factor‘ also hardly changed from the free Cu2+ ion value in all the methods listed
above. The MCSCF form factors for the two cases are compared in Fig. 6.2. The
trend of the energy changes in Table 6.1 is very similar to that of the free Cu2+
ion. Thus the crystal field as simulated by the point charge distribution has very

little effect on both the form factor and the total energy.

6.3 YBagCu305

The experimental situation regarding the ordered moment in YBagCuaOs is con-
troversial [kap194]. There are discrepancies between the moment values obtained
by two different groups, Refs. [jurg89, bur188] and Refs. [tran88L, tran88B], al-
though not so severe as in LagCuO4 ( as discussed in Sec. 6.4 ). The ordered
moment, )1, and TN in YBagCu306+, were found to be essentially constant for
0 S a: S 0.20 by both the groups [jurg89, tran88B], and the value of TN agreed
very well. The ordered moment )1 was found to be 0.64 :t 0.03113 for the single
crystal [jurg89] with a: = 0.0 and 0.15 , compared to 0.662t 0.07113, 0.46 21:0.05113,
and 0.50:1: 0.05113 for a: = —0.06, a: = —0.01, and 0.15 in Ref. [tranSSB]. Recently
a detailed experimental study of the neutron scattering form factor was reported
for YBagCugOms by Shamoto et. al.[sham93]. They found the Néel tempera-
ture to be 410 :1: 3K which is close to the value of TN for undoped YBagCugOs

[jurg89, tran88B, rebe89]. Since TN and 11 showed a linear dependence on each

88

 

 

 

 

10111111111111*111111111111
~ I l l l | -
- X); 3‘ _
: DUE] X x (1/21/2k):
[:1
0.6_— D g 3 —~
I 6 I
A 0.6 — E—
4 I q
3 — Q“ a -
11-1 - 11
0.4 :- (3/2 3/2 k) j
I D Cluster 3
0'2 _—+ Free ion __
C X Ion with point charg'ies
0.0 - lllllllllllllllllllllll-lllllq
0 1 2 3 4 5 6

41rsin6/A (.2171)

Figure 6.2: Comparison of form factor of free Cu“ ion, Cu2+ ion with point
charge distribution of YBagCuaog , and (Cu05)8" cluster form factor.

89

other and both were found to change very little from a: = 0.0 to a: = 0.15 for
YBagCu3Os+m one expects the form factor of YBagCugOms to be close to that
of YBagCugOg. Also, from Burlet et. al.’s experiment[bur188], it appears that the
relative form factor values at different reflections are insensitive to small changes
in the oxygen content. Therefore, as with LazCuO4, we will compare the shape
of the calculated magnetic form factor vs. momentum transfer, [(7] , with the
experiment even though the sample is not stoichiometric, since the absolute value

or overall scale factor is uncertain.

The structure of YBagCu3Oe is quite complicated as shown in Fig. 6.3. The
Cu2+ magnetic ions are antiferromagnetically coupled in the 2-dimensional C002
plane, but these Cu2+ ions are not at the center of inversion. Moreover the Cqu
plane has a double-layered structure. The spins are aligned in the C1102 plane but

the precise direction within the plane has not been determined [tran883, tran88L].
6.3.1 Cluster

We have chosen the six atom (Cu05)8‘ cluster in a tetragonal structure to repre-
sent YBagCu3Og. Similar clusters have been used by Sulaiman et. al.[sula90] for
calculating the hyperfine properties at the Cu“ site in this system. The cluster
electronic wave functions have been calculated using RHF and UHF procedures.
RHF and UHF calculations were performed with the COLUMBUS code[colu88]
and the Gaussian 92 code [g92] respectively, using contracted Gaussian basis sets.
Wachters’ [wach70] basis (13s9p5d) sets for Cu and Huzinaga [huzi71] basis sets
(956p) with additional diffuse p function for 0 were used. All 77 electrons in the
(CuOs)8' cluster were explicitly included in these ab initio calculations keeping
the total spin 5' = 1 / 2.

90

 

 

Figure 6.3: Crystal structure of YBagCuaoe . The contents of the (Cu05)8'
cluster are connected by the dotted lines.

91

Since we want to compare our calculated form factor with the experimental
results of Shamoto et. al. [sham93], we have used the values of the lattice param-
eters given by them to obtain the positions of the atoms in the cluster and the
environmental point charges. Therefore the Cu-O distance in the CuOg plane,
where there is a small buckling, was taken as 1.94121 and the distance between

Cu-O along the apical axis was taken to be 2.4421.

In addition to RHF and UHF calculations, we have also performed MCSCF
calculations using the Columbus code[coluSS] to investigate correlation effects. In
order to calculate the form factor, which is proportional to the Fourier transform
of the spin density, we have developed programs to obtain the natural orbitals

associated with spin up and down electrons separately ( see Sec. 4.2.3 ).

In the MCSCF calculations for the (Cu05)8‘ cluster, each doubly occupied
d-dominant orbital was correlated with one virtual orbital. This yields 354 con-
figurations. In principle, MCSCF[shep88] allows the d electrons in Cu to correlate
amongst themselves and.with the p electrons in the ligand. It also allows p — p
correlation within the Iigands. However, the result of this correlation calculations

suggests that d - d correlations are most important.
6.3.2 Environment

We took 8 unit cells for YB32C11303 as the environment, which gave 516 point
charges around the (Cu05)8' cluster. For a more realistic environment, the point
charges which had originally represented the nearest neighboring 4 Y3+’s and 4
Ba2+’s were replaced by effective core potentials (ECP)[wadt85]. The UHF result
of this procedure was reported in earlier work in Ref. [kap194]. Here we extended

our ECP by replacing the neighboring 4 Cu2+ ions in the 01102 plane by Cu“ ECP
92

[mart]. Introducing these ECP’s enabled us to incorporate Pauli repulsion effect
particularly between electrons in the 02" ions of the cluster and those belonging

to the neighboring extra-cluster ions.

6.3.3 Calculated form factor

The calculated value of the form factor for (Cu05)8‘ , according to Eq. 4.36 in
Sec. 4.3, is not a real number anymore because of the lack of inversion symmetry
of the cluster wave function itself. In order to compare the calculated form factor
with the experiment, we have to recover the inversion symmetry of the form factor.
To recover the inversion symmetry of the system, we consider two nearby Cqu
planes which form a bilayer structure. Now, the Y“ ion in the middle of the two

layers is a center of the symmetry as shown in Fig. 6.4.

We can write the spin density, fi,(F), associated with two Cu2+ ions, C111 and

Cu2 ( see Fig. 6.4 ), which has anti-inversion symmetry with respect to the central

Y.

Mr“) = pm — p(—F+2R') (6.1)

where p0") and p(—F+ 2R) are the cluster spin densities associated with Cul and
Cu2 respectively and R is the vector connecting Cul and central Y. The minus
sign between them indicates the spins of Cul and Cu2 are antiferromagnetically

ordered. The full form factor is f((j') which is the Fourier transform of [3,(1’).
f(é‘) = ] (pm — p(—F+ 2a) 12“?“de (6.2)
= 2iei‘f'fllm (fc((}’)e"f'n)

where fc((j') = f p(f')e‘f'fdf", the complex form factor which we can calculate from

our cluster wave function of (Cu05)8’ associated with a single ion, Cul.

93

Cu2

 

 

 

 

 

 

Figure 6.4: Bilayer structure of Cqu planes in YBaQCU303 . The small shadowed
circle is Y”. The big circles are Cu” ions ( the solid and empty circles represent
antiparallel spins). ‘

94

An observable quantity is the absolute value of f((j), which is given as
lfml = 2 11m (amerr’m (6.3)

In the experiment, Shamoto et. al.[sham93] determined the magnetic structure
factor F M from the integrated intensities of the magnetic Bragg peaks. From their

equation, the square of FM is proportional to

IFMI2 0< #7200931?) (6-4)

where they call 11 the magnetic moment, f ((1') is the magnetic form factor, and
51(5) is a bilayer structure factor, g(c}') = 2sin((i - R). Because Shamoto et. al.
took the form factor as the Cu2+ ionic form factor, their form factor is real and
does not include any covalence effect. When we compare our calculation with
the experiment, we have to compare |f((i')| in Eq. (6.3) with the corresponding

experimental value which is given as

11ml = 21(olsz'n1r- ml. (6.5)

From Eqs. (6.3) and (6.5), the calculated form factor, to be compared with

experiment, can be written as

= Ilm (fc<1>e“°‘°”)|
lsin(é'- R):

 

f(i) (6-6)

This is a form factor associated with one Cu2+ ion.

The calculated form factors from RHF, UHF and MCSCF (CuOs)‘8 cluster
spin densities are compared in Fig. 6.5-(a),(b). In Fig. 6.5-(a), the UHF form
factor is flatter than that of RHF in the small I6] region and the UHF form factor
lies above the RHF form factor at large M]. These trends of RHF-UHF form factor

differences also occurred in the previous cluster calculations on nickel compounds

95

Table 6.2: Comparison of total energies (eV) of clusters

 

 

(Cu05)8‘ in YBagCugos (Cu06)1°' in LagCuO4

 

 

 

RHF 0.0 ° 0.0 b

UHF -0.042 -0.044

MCSCF —2.693 -2.655
=&

 

 

 

“RHF energy = -2257.23425980 (a.u.) ( l a.u.=27.2ll4 eV )
”RHF energy = -2389.44155671 (a.u.)

discussed in Chap. 5. Considering the spin density in the F-space, we found more
spin density on the 0 sites in the CUOz plane in UHF than in RHF. This explains
why the UHF is flatter at small If]: more spin density on the 0’s in the Cqu
plane is canceled out due to AF ordering. As in the Cu2+ case, the difference at

large q is mainly due to core polarization.

In Fig. 6.5-(b), we can see that the MCSCF form factor differs little from the
UHF result. The UHF values lie above the MCSCF values except at the first few
Bragg peaks. This is similar to the result in the Cu“ ion as seen in Fig. 6.1-
(b). However the difference between UHF and MCSCF is slightly larger in the
cluster than in the free ion. It seems to indicate that existence of some correlation
(although very small) between d electrons in Cu and p electrons in O which does
not exist in the free ion. We also found less spin density at the 0 sites in the plane
in MCSCF compared to UHF. It makes the MCSCF values at small Itfl similar to
the RHF values. Our calculation suggests that the correlation effect makes the

spin density more localized near the Cu site.

The total energy values of RHF, UHF and MCSCF for the cluster are listed

in Table 6.2. It shows that the correlation energy of the cluster in MCSCF is

96

 

 

 

 

 

1.0 ' ' T ' r ' T ' fl r v v r I v v” v v r 1 I v v 1 r If v , , l f
(1/2 1/2 11) 3- 1
)- n¥¥ «- Wnu ‘
0.8 i- X "" a .1
. +X a
. +x : 6 .
1. "l' w j
- X .. 5 ,
A 114‘. x .. 116
CF ' + ix *' a if” 1
v 0.4 - (3/2 3/2 k) + -- .1
~— * .. ,
p 1b +
- X UHF J» X UHF .
0-2. + RHF T u MCSCF 1
. (a) t (b) j
0.0411111111111111111.1.1.11111111111

0 2 4 6 o 2 4 ° 6
411sin0/A (A71)

Figure 6.5: The calculated YBagCu3Oe form factor with RHF, UHF and MCSCF
methods.

97

similar to that of free Cu“ ion in Table 6.1. The form factor for free Cu2+ hardly
changed from MCSCF to CISD2; thus one might expect that the form factor of
CISD for the cluster is similar to that of MCSCF. Accordingly, we expected our
MCSCF to be a good approximation to find the correlation effect on the form
factor without doing extended CI calculations for the cluster. However a recent
work on the cluster of L32CUO4 by Martin and Hay [mart93] using extended CI
calculations contradicts this expectation: it gave the Mulliken charge values which
were quite different from RHF results. This will be discussed further at the end

of this chapter.

We compare the MCSCF form factors for the cluster, the free ion and the ion
with point charge distributions in Fig. 6.2. The cluster form factors are certainly
lower than the ion form factors at small IciI, the slope of the cluster form factor
being flatter than that for the ion. This is a manifestation of the covalence. In
the cluster, the electrons in O are allowed to hop to the Cu sites. It means
that spin density can be transferred from Cu to O. When an O is shared by
two antiferromagnetically ordered Cu’s, the transferred spin density at the 0 site
doesn’t give any contribution to the form factor at the AF wave vectors. It reduces
the form factor values at small Icfl and also leads to the covalent reduction of the
magnetic moment at the Cu site. However, at large I4], the covalent effect is hardly
seen, especially in the 2nd family (3/ 2, 3/ 2, 11:). It is because the form factor at
large I4] probes the spin density at small IFI, so it is insensitive to the spin density

near the 0 sites. '

6.3.4 Comparison with experiment

98

 

 

 

1.2 VfYYrY’rTTIV PVVI'VYVV‘JY'VVTVY'Y‘PVVV'I'VV'erYVIV‘VVTlvVVVIV'Yr‘
. .: I :
" 4» <- '1
1.0 e .1. ~
. x I i
. x I, L 9 I
o I' +$ D g 4‘ db 1
.6 _ $ 4- I "

b db

 

 

: :: «>1 3
30.6 r (1/2 1/2 k) If:- L 1
L, .

t :: ¥ S
0.4 " '1-

 

 

 

 

: 0 Cluster 1: (3/2 3/2 1:):
I * Experiment :2 :
0.0 ’....1....1....1....1..1.1...."1...1....1 ........ 1....111. ‘

 

l 1
0 1 2 3 4 5 0 1 2 3 4' 5 6
41rsin0/A (21")

Figure 6.6: The calculated ionic and cluster form factors were compared with
experiment [sham93] in YBanU3OQ .

99

We compare the MCSCF form factors with the experiments for YBagCu3Os in
Fig. 6.6. Because of the uncertainty in determining the absolute values of the
ordered moments in the present experimental results as mentioned in the intro-
duction part of this section, we again compare the shape of the form factor rather
than the absolute values. Thus the experimental values are scaled to give the
best fit to the calculated MCSCF values. Fig. 6.6 compares the MCSCF form
factor for the ion and the cluster with the experiment of Shamoto et. al.[sham93].
From Fig. 6.6, we see that the shape of the calculated cluster form factor agrees
with experiment except at the few points which have large error bars for both the
families, (1/2,1/2, k) and (3/2,3/2, k). When we compare the ionic form factor
with the experiment as Shamoto et. al. [sham93] did, it also gives a reasonable
agreement. Even though we see the covalence effect by comparing the ionic and
cluster form factor, the difference between them is within experimental error. In
order to see the covalence effect in the form factor, a more accurate experiment is

needed, at least according to this calculation.

From the experimental and theoretical study of the two families of Bragg
peaks, (1/2, 1/2, 11:) and (3/2,3/2, k), one sees that the slope of the form factor of
a given family is nearly flat. These small changes in a family indicate that the
spin density is mainly confined to the CuOg plane. This is confirmed by studying
the spin density of the cluster in the real space, F. Also, we found a very small
amount of negative spin density on the apical O site. This negative spin density
can be understood as the result of the exchange interaction between the oxygen p
electrons and the unpaired d electron (with positive spin) in the UHF and MCSCF
methods . (The result follows from the fact that the exchange interaction, which

is attractive, occurs only between parallel-spin electrons.)

100

6.4 LagCuO4

Even though LagCuO4 has been extensively studied since La2CuO4 was known as
a parent of the high Tc superconductor, the experiment on La2CuO4 still has un-
certainty, since preparation of a pure stoichiometric sample of LagCuO4 is difficult
and observable physical quantities such Néel temperature and structure transition
temperature strongly depend on the oxygen content. LagCuO4 is isostructural to
LagNiO4 , shown in Fig. 5.6. It is known that LanUO4 is tetragonal at high
temperatures and undergoes an orthorhombic distortion at lower temperature.
The transition temperature varies from 450K to 530K depending on the oxygen

vacancies (y) in LagCuO4-y [vakn87].

Different values of the Néel temperature, TN, and the ordered moment, )1,
were reported in different works: 185K and 0.30 [13 [fre188], and 250K and 0.40
pa Iyang87] respectively. Yamada et. al. measured TN and the ordered moments
on samples of LagCuO..-” with different y values; they found that higher TN
corresponded to higher’moment and reported 289K and 0.60 113 as the maximum
values of these two quantities in a particular samplerama87I. Finally, Keimer et.
al. reported a Néel temperature of 325K in their sample of LagCuO4, believed to
be very close to stoichiometric [keim92]. From Yamada et al.’s results one expects
this sample to have a larger )1 (unfortunately Keimer et al. didn’t measure the

ordered moment in their samples).

The experimental form factor measurement for more than one Bragg peak was
reported by only two groups[frel88, vakn87], as far as we know. Other experiments
measured only one Bragg peak to determine the ordered moment by using the

form factor of K2011F4 as discussed in Chap. 1. However their samples seem far

101

from stoichiometric judging from their low Néel temperature; 187K in Freltoft
et. al.[fre188] and 220K in Vaknin et. al.[vakn87]. Nevertheless, we will compare
our results with the later experiment of Freltoft et al.[frelSS], in the hope that the
shape might not be sensitive to stoichiometry. Some indication, that this might be
true, was given Burlet et. al.’s experiment in YBa.2CU306 [bur188I. Another form
factor measurement in a powder sample by Vaknin et. al.[vakn87], was claimed to
be a preliminary result by Freltoft et. al.[fre188I who measured the form factor in a
single crystal sample later. Thus we compare our result with the later experiment,

those of Freltoft et. al. [frel88].
6.4.1 Cluster

The cluster electronic wave functions for a (Cu06)1°“ in LagCuO4 have been
calculated using RHF, UHF and MCSCF procedures. We took our cluster as a
Cu ion and 6 ligand 0 ions in the appropriate tetragonal geometry with distances
between Cu-O as 1.89A‘in the Cqu plane and 2.41121 along the apical axis. We
ignored small orthorhombic distortion in our cluster calculation since we found

that a small distortion hardly changed the wave functions.

RHF and UHF calculations were performed with the COLUMBUS
code[colu88] and the Gaussian 92 code [g92] respectively, with the same basis
sets in YBanll305 calculation. All the 87 electrons of the (Cqu)1°' cluster,

considering S = 1/2, were included explicitly in the ab initio calculations.

To investigate correlation effects beyond HF, an MCSCF calculation with
COLUMBUSIcolu88] was performed for the (Cqu)‘°' cluster similarly to
YBa2Cu303 . The result of MCSCF with 354 configuration state functions in

(Cqu)l°‘ indicated that d-d correlation was most dominant, similar to the re-

102

sult in MCSCF for YBagCugOe .
6.4.2 Environment

We took a 552 point-charge environment outside the (Cu06)1°‘ cluster. The
positions of the point charges were determined from Cava et. al. IcavaSTI. The
point charge environment is PC333 corresponding to 27 unit cells, the same as in
LagNiO4 in Sec. 5.4. Beyond the simple point charge model, we first replaced the
neighboring point charges by ECP’stadt85I. The procedure of replacing point
charges by ECP’s in LagCuO4 is the same as in LagNiO4 ; denoted as ECPl (8
La” ECP’s), ECP2 (10 La3+ ECP’s) and ECP3 ( 10 La“ and 4 Cu2+ ECP’s).

6.4.3 Calculated form factor

The theoretical form factors were calculated from RHF, UHF and MCSCF for the
two families of the Bragg peaks in LagCuO4, (1/2,1/2, k) and (3/2, 3/2, k), similar
to YBagCugOg in tetragonal notation. The calculated form factors in Fig. 6.7-
(a),(b) show the covalent effect as a plateau at small chI. The overall changes from
RHF to UHF in Fig. 6.7-(a) and from UHF to MCSCF in Fig. 6.7-(b) are very

similar to the results in YBagCugOe.

The total energy values of RHF, UHF and MCSCF are listed in Table 6.2 and
the energy differences are also similar to those of YBagCugOs. The theoretically
calculated form factors of La2CuO4 and YBagCu3Og were found to be quite similar
despite a rather big difference between the basic clusters (one of the apical O’s
is missing in YBagCu303 ). We can understand the similarity of the results by
realizing that the spin density originates, mainly, from a d(,2-y2) orbital, which is

rather disconnected from the apical 0’3.

103

 

 

 

 

 

1.0 .s..,....,....,..‘_....,....,....,,
0.8; *X (1/21/21‘) _‘_ BE 6
* x 5
+ X ..
0.6- _—
- >¢< I a
A - +1.:x ._ as
a ' +X "' a
:04: (3/2 3/2 1;) + .1.
02- XUHF I XUHF
'. +RHF ‘2." DMCSCF
. (a) ‘t (b)
0.0....1....1....1 ,|,..‘|‘._ll“

o 2 4 6. .0. . . 2 4 6
41rsin6/A (ft-1)

Figure 6.7: The calculated L32C1104 form factor with RHF, UHF and MCSCF
methods.

104

For LagCuO4, we found the form factor changed when we replaced nearby
point charges by ECP’s, by roughly 5 %. Replacing a positive point charge by
a corresponding ECP seems to suppress the spin density of the nearby O sites
because ECP for a positive ion is less attractive to the electrons in the O’s than
the positive point charge. This effect is noticeable especially when two +3 point
charges at the positions of La3+’s right above and below the apical oxygens of the
cluster of LagCuO4 were replaced by La“ ECPS. In our paperIkapl94I, where we
included only the nearest La3+ ECP’s (of which there are 8) (ECPI), we had found
more spin density in the apical oxygen sites than in ECP3. The effect of reducing
the spin density at the oxygen sites by additional ECP’s was also found when
we replaced 4 more point charges by Cu2+ ECP in the CU02 plane in L32CUO4
and later in YB32C11306. However, the Cu2+ ECP replacement caused very little
change in the spin density. The general suppression of the spin density at the
0 sites resulted in smaller covalent reduction of theiordered moment defined in
Sec. 2.4. In addition to the effect from ECP, MCSCF also resulted in reducing

the spin density at the O sites in the Cqu plane, from the UHF- result.
6.4.4 Comparison with experiment

For 143201104, we compare the MCSCF form factors for the ion and the cluster with
Freltoft et. al.’s experiment[frelSS]. Because of the uncertainty in determining
the absolute values of the ordered moments in the present experimental results
for L32C1104 as mentioned in Sec. 6.4, we compared the shape of the form factor.
Thus the experimental values are scaled to give the best fit to the calculated
MCSCF values. One should note that the notation in Ref. [frelSSI for the Bragg

peaks is different from our tetragonal notation'because they used an orthorhombic

105

conventional notation. Fig. 6.8 shows that the cluster form factor agrees better
with the experiment than the ionic form factor. The form factor of LazCuO4 also

shows the plateau in the experiment as well as the cluster calculation.
6.5 SI‘QCUOQCIQ

We applied our cluster method to szCUOzClz , which has a tetragonal structure,
similar to La2CuO4 , but the LaO layers in LazCuO4 are replaced by SrCl. Since
the apical oxygens in LagCuO4 are replaced by C1, the study of the form factor of

Sl'zCUOzClz can give us some insight into the magnetic moment of the out-of—plane

oxygen sites,

Sl'zCLlOzClg is known to be antiferromagnetically ordered at ~ 250K and the

spins are aligned along the line connecting next nearest neighbor Cu-Cu as found

by Vaknin et. al.[vakn90].

The experimental form factor was measured by two groups, Ref. [wang90]
and Ref. [vakn90], and their experimental form factor agreed with each other
within experimental error. They measured the form factor at 15KIwang90I and
10K[vakn90I where SrgCuOgClg still has a tetragonal structure while LagCuO4
transforms from the tetragonal to orthorhombic structure below 540 K. The ex-
perimentally measured form factor of SrgCuOgClg by both the groups is similar to
that of LagCuO4 [fre188I. Wang et. al. [wang90] compared their experiment with
the Cu“ ion form factor and with band theory calculations, but neither of them
agreed with experiment. Later in this chapter, we compare our cluster calculation
with Wang et. al.’s to see the difference between their band calculation and our

cluster calculation of the form factor.

106

 

1.0

-
-
-
—
.-
-1
_1
-
.1
c-I
.1
-
-

0.6

0.6

f(QA)

0.4

llfllllljllllrii

I 1 Experiment
0.2 E—X Ion

-

.. U Cluster

0.0 l l l l I l l l l I l J 1 1 l l
0 2 4

411sin6/ )1 (24-1)

1111l1111l11L1l1111l1111

 

_

 

 

03"

Figure 6.8: The calculated ionic and cluster form factors were compared with
experiment [frel88] in LazCuO4 .

107

6.5.1 Cluster

The (CuO4C12)8‘cluster is similar to the (CuOs)1°' cluster except the two apical
oxygens in (Cu05)1°‘ are replaced by Cl+ ’s. The distance between Cu-O in the
Cqu plane is 1.9934 and the distance between Cu-Cl along the apical axis is
2.8614. The same basis sets for Cu, (1359s5d)[wach70I, and for O, (9s6p)[huzi71],
as in (Cu06)1°‘ were used for (CuO4C12)8‘. The basis set for C1 was also taken
form Ref. [huzi71] with an additional diffuse p orbital as (956p). All 103 electrons

in (CuO4C12)8"were treated via UHF calculations using Gaussian 92 code[g92].
6.5.2 Environment

The point charges within 27 unit cells were taken via Evjen’s method discussed
in in Sec. 4.4.2, as in 143201.104 . The positions of point charges were determined
from Vaknin et. al. [vakn90]. Beyond the point charge model, the same ECP
environment, ECP3 ( 10 Sr2+’s ECP and 4 Cu“ ’3 ECP), as in LagCuO4 was
taken except that the La3+ ECP in LagCuO4 was replaced by the Sr2+ ECP

[wadt85].
6.5.3 Comparison with experiment

The calculated UHF form factors are compared with the experimental results of
Wang et. al.[wang90] and the ionic form factor in Fig. 6.9. The UHF‘result
reproduced the plateau in the small q region as seen in the experiment [wang90].
Compared to the band calculation (LDA) carried out by Wang et. al.[wang90],
or the free Cu“ ion calculation, our ab initio cluster results seem to agree much

better with the experiment, thus providing a good justification of our method.

108

 

1.0 Illllllflllll lllFIlITlIlll

Xxx

11*

[Ill

 

 

 

 

0.8_ -_I

0.6r El ...,

A I. .1

<1 _ x _

G - -l
v

'11-! r— . .1

0.4— _-

: 9 3

.. f Experlment .

3 U Cluster :

(LOP-llllllllll1111l1111l1111l111

0 1 2 3 4 5 6

4Trsin0 / >1 (ll-1)

Figure 6.9: The calculated ionic and cluster form factors were compared with
experiment[wang90] in StaCUOzClz .

109

Table 6.3: Covalence factors obtained from the cluster calculations defined in

Eq. (2.68).

 

 

MCSCF (ECP3) UHF (ECP3) UHF (ECPl)

 

YBagCu306 0.85 0.81 0.78
LagCuO4 0.84 0.80 0.75

 

 

 

 

 

The calculated and experimental form factors of Sl‘gCUO-zc 12 are similar to
that in LazCuO4 , showing a plateau in the small q region. It indicates that
replacing the apical oxygen by C1 hardly affects the spin density in general and
in particular the out of plane spin density is negligible. The spin density in the

layered cuprate materials seems to be primarily confined in the Cqu plane.

6.6 Conclusions

From the study of the form factor of YBagCugOs, LazCuO4, and szCUOzClg
, we conclude that the calculated shapes of the neutron form factor agree rea-
sonably well with experiments for these cuprate materials. For La2CuO4 and
SI'2C1102CI2 , the cluster form factor agrees with the experiments much better
than the ionic form factor. Within a family of (i values when only q, increases,
the slope is nearly flat. That lead us to conclude that the spin density of the
cuprate material is confined in mainly the Cqu plane. We have also improved
our calculated results including correlation effects via MCSCF and introducing
additional ECP’s. Unfortunately, the changes in the form factor by these efforts

lie within the experimental errors.

The cluster results when compared to the ionic form factor clearly shows the

110

covalence effect. The covalence factors, calculated by using Eq.( 2.68) in Sec. 2.4,
are listed in the Table 6.3. The covalence factor is sensitive to the EC P environ-
ment because replacing the point charges by ECP, changes the spin density at the
planar oxygen as well as at the apical oxygen. The covalence factor is determined
by how much spin density is at the planar oxygen sites since the spin density
at these oxygen sites is canceled by AF ordering. We also found that the cova-
lence factor has changed by including correlation effects via MCSCF. However.
our MCSCF calculations, which are limited to 354 configuration state functions,
might not be good enough to give accurately the effect of correlation on them-
valence factor. MCSCF gave less covalence than UHF, which is anti-intuitive,i.e.
we expected that including correlation should lead to more covalence. Moreover
from an analysis of the Mulliken charge, Martin and Hay [mart93] found more
covalence in their CI calculation compared to our MCSCF, while our RHF results
agree with theirs. In order to study the covalence effect more accurately, we have
to expand our MCSCF calculation to include more configuration state functions.

This will be a next step following this thesis work.

The ab initio cluster method appears to work rather well in the form factor
calculation for the cuprate system. Then the question is, why the same method
failed so badly to reproduce the experimental form factors for LagNiO4 , which is
isostructural to La2CuO4 . The main difference between L32 N iO4 and LagCuO4 is
the different spin states of Ni“ and Cu2+ . In the ground state of the (NiOs)1°"
cluster, two molecular orbitals, which have eg symmetry such as drz-yz and d322_,2 ,
are singly occupied. Thus the spin density in (Ni05)1°’ is 3-dimensional while the
spin density in (Cqu)1°' is 2-dimensional. This explains why replacing apical

La3+ point charges by ECP changed the form factor values in LagNiO4 , by about

111

10%, more compared to L32CUO4 where this change is about 5%). Unfortunately
there is only one measurement of the form factor for LazNiO4 which showed a
dramatic flattening, i.e. a big plateau in the small Icfl region, which was not seen
in the form factor of other nickel compounds such as KNiF3 and NiO. We have
investigated several possibilities to explain the observed plateau, such as a mixing
of the singlet and triplet spin states on each siteIkap194I and other configurations
for the ground state such as one tgg and one e9 state are singly occupied rather
than the two eg’s. But these attempts failed to reproduce the observed shape of
the form factor. An extensive correlation calculation on the cluster might resolve
this problem. Also another experiment is needed to confirm the observed strange

form factor seen in LagNiO4 [wang91].

Appendix A

Perturbation calculation of the
contribution of the 4-spin
hamiltonian 71(4)

H“) can be written in terms of spin operators [taka77]

1 “ " :1. '1
H“) = Us { Z4130 — 45.- ~51) + Eff-”(4b.- - bk -1)

1'<j 1<k

+ Z tijtjktkltli [8061' ° Sj)(51. - 5;)

loop

+80(§,- - §,)(§, . s.) — 80(§,- - 5,.)(5‘, . s.

V

_4(5,.§,)_4(§,.5,) +11 } (M)

If we take the spin-up sublattice, the spin operators in the Holstein-Primakofl

representation are given by

 

al-a- 1/2

5;. = 1/2—s'(1— 215’) 11,, (A2)
at... "’

S, = ajx/é?( —7’31) (13)

5‘ = S—aja,. (A4)

J

In spin wave approximation [matt6-5. kitt63I, the above spin operators can be

expanded in powers of (1/ S ) as

1
1.} z \/2—S(1—(-§)a}aj)aj,

1
5, 3 Ma} (1 — (E) aIaj). (A.5)

We introduce the spin operators in 1: space as

1 41311
J /N g k
1 Z -"

We then use Bogoliubov transformations which makes Ho diagonal, i.e.,

a; —+ (coshu;) B; + (sinh 11;) Big»

a; —+ (cosh 11;) B; + (sinh 11;) B_;, (A.7)

with u; = u_; = 11*; and requiring the condition, tanh 211; = (l/Z) Z, cos(h - 7').

Keeping the leading terms of 0(1/5) inside the square brakets, 71,, reduces to

12 2 1
H, = (1(7) Eg- [-N(1+ E) +%Z(n; +1),/1— 7,3 (A.8)
r

where n; = BEE; and 7;, = (l/Z) flied".

H“) can be separated into two parts, diagonal terms and off-diagonal terms,

71(4) = V1111; + Voff (A.9)

114

where

t4 fi 1
Vdia = — [3N+48232 { ;V(1+ E) _

U3 I 1‘72}

(\DIH

2:071; +

E

rle [0

+852 {N - £84011; + c(/3))}
E
1 .. ..
+852 {N - 3,- Z(f(k)n,-; + WM
1?

9 4
+805“{2( N (1+§)—32(n;+ ) 1—72)

I?

[Cur-

—(N—::Z(a( (i?)n,;+c(i)))}], (4.10)

and

v,” =4522—{(b(12)+g(.))(B,§B:,;+B_,;B,;)}

+8054§3 { “mags; + B_;B;)} ]- (A.11)
For 2-dimensional system, relevant to our cuprate and related systems,
a(h) = 2(pi+uz)(1 —cosh-?zcosh-1"y),
b(h) = 211111;,(1 — cos h- 1'", cos I: - fi),
15) = 21/: - 21:: cos h- 1"} cos h- r,,
I5) = (pi+u,3)(2—cos2lii- 1"; —cos2h-?y),
901:.) = pku1(2 — cos 2h - i": - cos 2}; - i’y),

h(h) = 21/: - pflcos 2h - 1‘"; + cos 2F - f'y), (A.12)

where the relation #i — u: = 1 holds.

115

The expectation value of the site spin operator, 5:, is given by
(5}) =(‘1’I5fl‘1’). (A-13)

where W is the ground state of the Hamiltonian in Eq. (2.58). W = \IJO+6\II, where
We is the ground state of ’HO, \Ilo = I0), i.e. the zero magnon state, and 6‘1! is the

perturbed eigenstate associated with H“), then we have

(4|) I
5111-.- EL— (0111 1'3”)

11' El)”

————’|u) (A.14)

where III’) are the two magnon states and Eo’s are the eigenvalues of 710. Here

(OIH(4)III’) has non-zero contribution from only the V0,; part of H“).

We obtain (5;) up to the first correction as
(5;) = (\IIOISflWo) + 2(‘IIOISjI6\Il) (A.15)

For the square lattice with S = 1/2, the two terms in Eq. (A.15) are given

(WOISJ‘ZI‘IIOI E (52)}{eis

 

1 1 1
— -Z(-) . (A.16)
N i; 2 1- 72
(‘11 IS’I6W) - (t—2-)-1— Z iISU - cosh - ‘F cosh-1")
° 1’ " U2 4N E (1_7£)3/2 I y
—(2 — c0321: - “1",, - c0321; - 1)], (A.17)

where 71,: l 22 e’k ”-1, —%(cosk rz+coslc Ty). We calculate the above summations
numerically. After checking the convergency of our numerical calculations, we
obtain (5,)Heg. = 0.30362 in Eq. (A.16) which agrees with Anderson’s well-known
result [ande52] and 2(WOISjl6W) = 1.2839(5) in Eq. (A.17).

116

Appendix B

Construction of natural orbitals

in MCSCF

In this appendix, I describe the development of the programs to construct natural

orbitals (NO) in our MCSCF calculation.

The one particle density matrix is given by Eq. (4.23)
7(1'|1) = N/¢'(1',2,....N)¢1(1,2,....N)dr(2, ...N). (8.1)

Eq. (B.1) can be written .With the determinants in the configuration state functions

(CSF)’s of Eq. (4.19) and Eq.( 4.20)

7(1’ll) = N Z 222 C1010;1011<D£ID{>2..~. (8.2)
I J I: 1

Let’s define I(I, J, k, l) as

[(Isjakal) (DilDlIIZ-N

= / Dg'Dfdr(2...N) (3.3)
Then [(1, J, k, I) can be reduced to a form of

1(1. J. k. 1) = 14,2 2 ¢:(1')a..(I. J. k. 0151(1). (3.4)

117

where 0;,(1, J, k, I) is determined from Eq. (B3). The matrix A,,- is

Agj == Z:ZZC;CJCZICUG;J(I,J,k,f) (8.5)
I J k I
In order to construct Ag], we have to find the non-zero contribution from

aij(Ia J9 k9 1)

In an MCSCF calculation, only active molecular orbitals, which form configu-
ration state functions, contribute non-diagonal terms in Agj, so we can construct
14,-,- whose dimension is (Nactive x Nactive) rather than (NMO x NMO). ’NMO’
is the total number of molecular orbitals and ’Nactive’ is the number of active

molecular orbitals.

3.1 Development of programs

The construction of N O’s consists of the following steps.
(1) Read the determinants, DI and D{.

The information of CSF including the determinants and the CI coefficients can

be read from the file generated by mcpc.x in COLUMBUS code[colu88].
(2) Construct the one particle density matrix Ag,- .

We can construct Ag,- in the basis of active molecular spin orbitals by finding

non-zero contribution of I (I , J, k, I)

A determinant, Di, for N active electrons with N active active molecular spin

orbitals, is written as

Di = |¢1(1)¢2(2)m¢.(v)~-.¢1(N)I (8.6)
where u is the index for an electron and i, k are the indexes for molecular spin

orbitals running between 1 and N active.

118

When DI and D}, are the same. I(I.J, k. 1) becomes

OCC

I,,,(IJkl)=—Z¢'(1,- (1) (B?)

The other non-zero contribution to [(1, J. k. 1) comes only from the pair of deter-

minants;

Di |¢1(1)¢2(‘2).-.¢.~(v)....<z>1(NH
01’ = |¢1(1)<!>2(2).-.<b1(p)....¢1(N)I (8.8)

when i =j but u #11, or when i aéj with 11:11. Then

I(I.J,k,l) = $(— 1W“) ¢:(1’)¢,(1) for #1
1
= N¢:(1’)¢.(1) for men (8.9)

After summation on (I,J.k,l) in Eq. (3.5) and changing dummy indexes 11,}! to i,j

we obtain the one-particle density matrix for active space

Nactive Nactwe

(1’=|1)Z Z a);( (1'()a)1 (8.10)

( 3) Diagonalize 14,-, and find eigenvalues and eigenvectors.

Eq. (B.10) can be written with a vector Q = (49162 ..... (by) and a matrix A =

AU
7(1’I1)=<I>(1’)A<I>*(1) (B.11)
If we chose a proper unitary matrix, C and it = <I>C, then 7(1’I1) becomes
(1'(1): <i>1(')C*AC<i>*(1) (8.12)
where

C'AC = N (8.13)
119

where N is a diagonal matrix whose elements. 12. is. are called occupancy numbers.

Eq. (3.13) is reduced to the eigenvalue problem

AC = ON. (8.14)

(4) Construct NO's from the eigenvectors.

By solving the eigenvalue problem of the Eq. (8.14), we can construct NO’s

by (i = QC from the active MO’s.

In the procedure (3), Aij can be separated to 0,1 and 13,-] for up-spin orbitals
and down-spin orbital respectively because there is no spin cross term in Ag,»
Then we can construct a-NO’s and fl-NO’s separately in the procedure (4). From

a-NO’s and B-NO’s, we can calculate the spin density.

The above procedures are integrated in the main program usually named as
norb-***.for with the subroutine program, density.sub.for. Fig. B.1 shows

the structure of the programs.

 

norb
(main program)
input: readmcpc.inp
input: mocoef.dat

I
l l

 

 

 

 

 

 

 

 

 

 

 

 

density_matrix eigensolution
(subroutine) (subroutine)
construct matrixA solve eigenvalue problem
I
tred2,tqli,pythag
(subroutines)
matrix diagonalization

 

 

 

Figure 3.1: Structure of program sets to construct NO's

1‘21

Appendix C

Failure of the Mulliken Charge
Population Analysis

In the UHF calculations of NiO and KNlF3, we found the Mulliken charge
values[levi91] obtained with diffuse basis functions in Ni (14sllp6d) to be quite
different from the nominal charge values for the ionic material (see Table C.1).
Particularly for NiO, the discrepancy is very large. These values (-0.21 for Ni and
-1.63 for O) for NiO seem to contradict the assumption that NiO is highly ionic,
which allows us to assign the nominal point charge values, +2 for Ni and -2 for
0, when generating the Madelung potential. Therefore we performed the same
cluster UHF calculations without diffuse basis functions in Ni (1339p5d) to see
how sensitive the Mulliken charge population was to the choice of basis functions.
The Mulliken charge values without diffuse basis functions (1359p5d) were found
to be +1.75 for Ni and -—1.95 for 0, much closer to the nominal point charge
values. (Similar values were obtained by Sulaiman et. al. [Sah090] in their cluster

calculations of YBagCu303 and LazCuO4).

Even though the results of the two calculations, with different basis sets, look so

different in Mulliken charge analysis, we found that physically meaningful quan-

12?.

Table (3.1: Mulliken charge population values for different basis sets for Ni

 

 

NiO KNiF3

 

Ni 0 Ni F

 

(l4sllp6d) cm -1.63 1.76 -0.96
(13s9p5d) 1.75 -l.96 1.86 ~0.98

4
4

 

 

 

 

 

 

 

 

tities, such as the charge and the spin density, are essentially the same. Our
understanding of how this is possible is the following. The diffuse functions in
Ni (14sllp6d) are so diffuse that they spread over the neighboring oxygen sites
and can mimic the diffuse function on the oxygens. We estimate that the approx-
imately 1/3 of an electron per 0 assigned to the diffuse Ni orbitals are physically

associated with the oxygen ions.

The Mulliken population values in KNiF3 are rather stable with respect to
the choice of basis sets and the calculated values are close to their ionic charges
(+2 for Ni2+ and —1 for F ") even when we use the diffuse basis functions in Ni (
see Table C.1 ). This is due to the fact that the F’ wave function is much more
compact compared with the 02‘ wave function so that the diffuse Ni functions

are simply not appreciably occupied.

In summary, these results show that the Mulliken charges assigned to different
ions (such as Ni,F,O etc.) depend not only on the choice of basis set but also on
the type of ions in the cluster. (A similar problem was noted by Noell[noe182] and
Bauschlicher and Bagus[baus84] for the transition metal complexes.) Therefore
it sometimes may be misleading to use these charge assignments in describing

physical quantities such as the electrostatic potential. As we just noted, our

123

NiO results give an extreme example: clearly the assignment to Ni of electrons
in orbitals centered on Ni but so diffuse that most of their weight is at the Ni-O
distance, is not sensible. The Mulliken assignment is much more sensible when the
orbitals are not so diffuse. The charge density was in fact found to be essentially
the same with or without these diffuse Ni functions, so it is clearly reasonable to

prefer the Mulliken charges calculated without diffuse functions.

1‘24

Appendix D

Crystal field splitting in KNiF3
and NiO

In addition to the form factor, we have calculated the crystal field splitting in
KNiF3 and NiO in various calculation methods such as RHF, MCSCF, and CI.
In this appendix, we briefly discuss our cluster calculation results of the crystal

field splittings.

The atomic 3d orbitals of a transition metal ion like Ni2+ are degenerated if
it is isolated. However a Ni2+ ion in KN iF3 or NiO is subject to a crystal field of
octahedral symmetry. The atomic 3d orbitals of a metal ion split, in the presence of
such a field, into a triply degenerate tgg level and a doubly degenerate 69 level. The
129 orbitals are higher in energy than the tgg orbitals, and the difference between
them is called lODq or the crystal field splitting. This lODq value is equivalent to
the difference in one-electron energy between t'" e“ and tg’leg“. For N i2+ with

299

8 d-electrons, lODq is the energy difference between tggeg and tggeg. Within the

Hartree-Fock scheme for the cluster, lODq may be defined as the energy difference

between two independently calculated N-electron states, 31429 and 3ng [elli68]:

lODq = E3129 - EBA” (D.l)

125

Table D1: lODq values for I(NiFg and NiO (cm“)

 

 

[\IJV1F3 1V10

 

Experiment 6980 a 9114 b

 

RHF 5842 6363
MCSC F 6516 7114
CI 6224 6783

 

 

 

 

 

 

 

 

“Ref. [ferg64]
”Ref. [newm59]

Here 3A29 and 3‘ng are the ground state and the first excited state, respectively,

of the cluster, (NiF6)"’ or (Ni06)1°" .

RHF, MCSCF and CI calculations had been carried out for octahedral
(NiF6)" and (Ni06)1°‘ cluster with COLUMBUS[c011188] to get lODq. We note
that the (NiF6)"’ cluster calculations have been done with the nearest neighbor-
ing 8 K+ ECP and 474 point charge environment while the (Ni06)‘°‘ cluster

calculations done with only 722 point charge environment.

The lODq value for KNiF3 obtained by MCSCF agrees well with the experi-
mental value within ~ 7% error. This is the best agreement with the experimental
lODq using ab initio method for KNiF3 as far as we know. For NiO. our best num-

ber from MCSCF is about 22% off from the experimental value even though we

improved it from RHF to MCSCF.

1°26

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1