VARiATIONAL APPROACH TO AN
APPROXIMATE HAMILTONIAN FOR
STRONGLY CORRELATED NARROW-BAND
ELECTRONS - RELATION TO BAND
CALCULATIONS AND THEORY OF
EXCHANGE IN INSULATORS

Thesss for the Degree of Ph. D.
MITCHTGAN STATE UNIVERSITY
MILTON PENHA SILVA
1975

    
   

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VARIATIONAL APPROACH TO AN APPROXIMATE HAMILTONIAN FOR
STRONGLY CORRELATED NARROW-BAND ELECTRONS-RELATION TO BAND
CALCULATIONS AND THEORY OF EXCHANGE IN INSULATORS

presented by
NILTON PENHA SILVA

has been accepted towards fulfillment
of the requirements for

PH.D. Physics

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ABSTRACT

VARIATIONAL APPROACH TO AN APPROXIMATE HAMILTONIAN FOR
STRONGLY CORRELATED NARROW-BAND ELECTRONS - RELATION TO

BAND CALCULATIONS AND THEORY OF EXCHANGE IN INSULATORS

BY

Nilton Penha Silva

This paper contains a critique of one-electron band
theory for narrow-band magnetic semiconductors, with the
conclusion that the approaches in the literature are
unsatisfactory from a conceptual point of view, the
difficulty being connected with the question of what
periodic potential, if any, is to be considered. We
then present a new approach which is variational and
consists of taking a trial Hamiltonian H, much simpler
than the exact Hamiltonian, nevertheless containing
two-electron terms as well as one-electron terms. Unlike
one—electron theory, physical predictions are to be made
only when the one-electron or band terms in H are combined
with important two-electron terms. Unlike previous

theories (due to Hubbard, Anderson) which follow the

Nilton Penha Silva

latter procedure recognizing the essential importance of
certain two—particle terms, we choose the one-electron
terms (the band-structure) variationally and simultaneously
with a similar choice of the two-electron terms. This
novel feature of our approach overcomes the difficulties

in the previous theories, as shown by our initial investi-
gations made within the narrow half-filled single-band
problem. Two definitions of our trial Hamiltonian H are
considered; in one, H is of the form of the Hubbard
Hamiltonian, and in other, potential exchange terms are

added.

VARIATIONAL APPROACH TO AN APPROXIMATE HAMILTONIAN FOR
STRONGLY CORRELATED NARROW-BAND ELECTRONS - RELATION TO

BAND CALCULATIONS AND THEORY OF EXCHANGE IN INSULATORS

BY

Nilton Penha Silva

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1975

DEDICATION

In memory of my father.

ii

ACKNOWLEDGMENTS

I am deeply grateful to Professor Thomas A. Kaplan for
having me as his student and giving me the opportunity of
collaborating with him in physics research. His guidance
has been superb. Truly, without his enthusiasm, time,
effort, encouragement and friendship, the accomplishment
of this thesis would not have been possible.

I am very thankful to Professor Truman O. Woodruff for
accepting me as a student at this Department of Physics, for
being my academic advisor and for continuous encouragement.

I thank very much The Rockefeller Foundation for pro-
viding me with a scholarship through all this doctoral
program.

I am thankful to Professor Julius Kovacs, Professor
Wayne Repko and Professor Gerald Pollack for their encourage-
ment and continuous interest in my progress.

I am indebted to Professor Robert D. Spence and
Professor S. D. Mahanti for valuable comments on this work.

Last but not least, I thank my wife for her invaluable

assistance, encouragement, patience and support.

iii

LIST OF
LIST OF
Chapter

I.

II.

III.

IV.

V.

VI.

TABLE OF CONTENTS

Page
TABLES Vi
FIGURES Vii
INTRODUCTION 1
HUBBARD'S DERIVATION OF THE HUBBARD HAMILTONIAN;
ITS FAILURE 9
ANDERSON'S LIGAND FIELD THEORY; ITS UNACCEPTABILITY 16
VARIATIONAL APPROACH 26
EXPLORATION OF THE VARIATIONAL APPROACH WITHIN
THE SINGLE-BAND PROBLEM 31
A. Zero-Bandwidth Hamiltonian; All Temperatures 33
B. Narrow Half-Filled Band: The "Best" Hubbard
Hamiltonian 42
B.1. High-Temperature Limit 44
B.2. 'Low-Temperature Limit 48
3.3. Two Sites; All Temperatures 59
C. Narrow Half-Filled Band: The "Best" Modified
Hubbard Hamiltonian (With Potential Exchange) 64
C.l. Two Sites; Low-Temperature Limit 64
C.2. Effective One-Electron Potential 75
SUMMARY AND DISCUSSION 80

iv

APPENDICES

A1.

A2.

A3.
A4.
A5.

A6.

A7.

A8.

REFERENCES

Two-Site Single-Band Models

Natural Conditions for a Unique Choice of
Wannier Functions

Solutions of Equations (60) and (61)
Proof of Equality in Equation (92)
Solution of Equation (101)

Proof that fieff in (114) Corresponds to a
Heisenberg Hamiltonian

Calculation of Equation (121)

Calculation of Terms in Equation (127)

Page

84

92
94
99
101

105
107

110

112

Table 1.

Table 2.

Table 3.

LIST OF TABLES
Page
Listed are the eigenstates of the exact 89
Hamiltonian H, (138), and E - uN for
half-filled band condition; a is a function
of temperature and other parameters in H

(See Appendix A1.). Here,

_ 2 *2
((V V 'I’ 16t12) ] I

2
12) i 12)
- E-)/2t A = (1 + c2)”2
12'

1
I — _ _ .—

n = 3h + v + 2v12 - x12 , |a1> = (c

11
+ + + + + +
°2+°1+II°> ' Ia2> ‘ (°1+C1+ + °2+°2+)I°> '

E1 = (1/2) [(V+v

C = (V12

t = h

Mud ~

X12 '
+ +
1+°2+

12 12

+

~ +
.—=
Hx . Here E 91

2 2 S
2) i ((U - U12) + 16b12) ] ,
2

_~l- ~l = v
E )/2b12 and A (1 + C

The singlet eigenstates of H

(1/2)[(U + U

5—:

~ -1’
c = 2
C (012 ) .

Eigenstates of H1 and H1 - uNl . The last 98

column is for u = b11 + (half-filled band).

«HG

vi

Figure 1.

Figure 2.

LIST OF FIGURES
Page

The values of b U and b for the "best" 63

12' ll

half-filled Hubbard Hamiltonian are shown as
functions of kT, in units of V. Also is
shown the ratio blZ/U' The parameters in
the exact Hamiltonian were, in units of V:

v = 0.2, h

= ' =
12 0.05, V 0.03 (so

12 12

0.08), h11 = X12 = 0.

t12

The eigenvalues of i) H-uN, ii) HH - uN 74

and iii) HHX - uN in the zero and small

overlap limits for the half-filled-band

grand canonical ensemble. In case i) it is
I

assumed that t12 < 0, v12 < 0 and

|t12| Ivizl and kT << ||t12| - Ivizll .

The levels in ii) and iii) were drawn

roughly as in the best HH and HHX .

vii

I . INTRODUCTION

This paper contains a critique of one—electron band
theory for narrow-band magnetic semiconductors, e.g. NiO,
EuS, commonly spoken of as Mott insulators. The

3'
conclusion of this discussion is that the approachesl-6 in

KMnF

the literature are unsatisfactory from a conceptual point
of View. The difficulty we are discussing is not
connected with the solution of a given one-electron
periodic potential ("self-consistent" or not), but it is
rather concerned with the question of what periodic poten-
tial, if any, should be considered. We go on to present
an approach which is shown to have promise of being satis-
factory (it overcomes all the objections raised in our
critique).

We neglect phonon effects, focussing on the
Schr6dinger equation which describes the motion of the
electrons in the Coulomb field of nuclei fixed in space.
Nevertheless, this equation is not that of a single
electron moving in a fixed potential, because of the
Coulomb interaction between electrons — an electron "sees"
the fixed nuclei and all the other electrons at their
instantaneous positions: the Hamiltonian assumed is

P 2 I 2

+V(ri)] +§§ S- (1)

N
p.

H=Zlm
1

where V(r) is the potential due the nuclei. Hence the con-
cept of a one-electron theory - a model where the electrons
interact with some effective fixed potential Veff’ but not
with each other - by no means obviously flows from the
fundamental physics defined by (1). Nonetheless, this con-
cept forms the basis of much of solid-state, molecular and
atomic physics. A large portion of solid state work takes
the nuclear positions to form the crystal structure. The
resulting translational symmetry simplifies appreciably the
one-electron problem (given the periodic potential), but
does not bring the original problem (1) perceptibly closer
to explicit solution.

Historically, some theoretical foundation was given
to the one-electron theory in the variationally best
one-electron approximation, the Hartree-Fock Approximation

(HFA)7’8’9. However it was recognized early10

that the

common restricted version of this (plane wave solutions)

was poor for the low—temperature properties of the electron
gas, and presumably therefore for broad-band metals. The

HFA is, again for non-zero temperature, also extremely poor
and misleading for models of very narrow partially-filled-band

11-15 such as the Mott insulators in which we are

insulators
interested, even in its general (unrestricted) form.

The use of one-electron potentials other than the HF
EXJtential for the calculation of energy bands has been

Wiriespread. In the remaining paragraphs of this introductory

secztion we will discuss methods which were not specifically

designed for narrow-band situations and which either have
been applied to narrow-band materials in which we are
interested or appear to have possibilities in this direction.
In sections II and III, methods that were addressed explicitly
to narrow band electrons will be considered in some detail.
Finally in section IV we discuss our proposal as a contribu-
tion to the search for a satisfactory definition of
one-electron states in Mott insulators; and in section V our
approach is investigated for the half-filled-single-band
problem.

Probably the most widely used potential for band calcu-

1/3 exchange"1 or its well-known gener-

alization known as the "Xa" method16. Slater's exchange was

lations is Slater's "p

originally developed1 as a simplifying approximation to the
Hartree-Fock method. Namely the exchange terms were first
averaged and the average exchange was further simplified by

replacing the electron density p in the free-electron expres-

 

sion for the average exchange by p(r), the electron density
in the crystal. This leads to an exchange term -Aop(r)1/3
where A0 is a specific positive numerical constant. The
Xa method consists of replacing AG by a positive parameter A
which is unspecified theoretically. The one-electron
Hamiltonian G in this approach is the sum of three terms:

the kinetic energy, the electrostatic potential energy due to

an electron density p(r) plus the nuclei, and -Ap(r)1/3. The

final one-electron picture consists of one-electron states

w.

1.81 which satisfy Gwi=eiwi and the "self-consistency

requirement" p(r) = LIWiIIIIZr the sum taken over occupied
i

states. Later it was argued that Slater's approximation1

to the HFA would actually be better than the latter in
certain respects, and in particular for energy band

calculationsl7. Clearly the derivation of Slater's approxi-

 

mation as a method of improving on the HFA must be viewed
as an arbitrary procedure, and the Xa method has to be seen
as phenomenology.

Furthermore, the consequent uncertainty in the meaning
of these procedures does not decrease as one proceeds from
nearly-free-electron situations to cases of narrow bands18
(because of the dependence on the free-electron case of the
original argumentl). In this connection we point out that

1/3 schemes are unsatisfactory on intuitive grounds

these p
in connection with an N-electron atomic (or ionic) calcula-
tion. For large distance r from the nucleus, p(r)
approaches zero exponentially; therefore the total potential
in the X0 scheme does not approach the physically expected
potential due to the (N-l)-e1ectron ion (e.g. if the
N—electron atom is neutral, the Xa scheme gives a total
potential energy that is exponentially small at large r
rather than the expected -e2/r). We note that disagreement
with the same physical expectation was the basis of Slater's
criticism of the excited or unoccupied states in HF theory
(see footnote 15), and that he claimed17 a major improve-

1/3

ment for his p method in this connection. Clearly, a
failure to properly predict the large-r behavior in atoms
will introduce serious uncertainties into the prediction

of band-structure effects in narrow bands.

A more fundamental approach based on the real physical
problem of interest defined by (1) was taken by Kohn and
coworkers2 who developed a new variational principle and,
making certain approximations appropriate to either
slowly-varying p(r) or to p(r) with a large constant part,
derived a one-electron potential similar to Slater's
but with A = g-Ao. However, the authors2 noted that their
theory could not make justifiable predictions "at the
surface of atoms"; hence one would not expect band-structure
effects in narrow bands to be discussed satisfactorily
within this approach in accordance with the discussion of
the previous paragraph.

Another approach to one-electron states in crystals
which is not basically of a phenomenological nature is
the field-theoretical approach described by Hedin and
Lundqvist3. Here one-particle states (wave-functions fs(x)
and corresponding energies 88; x stands for both space
and spin coordinates) are precisely defined in principle in
terms of the exact many-electron energy eigenstates, and
it is shown that the one-electron Green's function (and
therefore many physically observable quantities) can be
determined in principle from these states. Unfortunately.
this definition does not appear to be appropriate in its
present form to the narrow-band regime, at least for
obtaining a complete orthonormal set of one-electron func-
tions, since these single-particle wave functions fs(x)

3,19

are not linearly independent for interacting electrons;

for our case of interest, where the interactions are very
strong, criteria for the choice of an appropriate subset
of the fs(x) (which further have to be normalized) do not
existzo.

The final band-theoretical method that was not origin-
ally presented in the context of narrow-band situations
is the Wigner-Seitz potentia14. This turns out to be
related to certain results of our variational calculation,
and will be discussed in section V.

Hubbard's derivation of the well-known Hubbard
Hamiltonian HH, which is presumed to be appropriate for
narrow-band systems is discussed in section II. It is
construed here as involving a one-electron band calcula-
tion in the sense that it involves a calculation of the

one-electron operator appearing in H Anderson's "ligand

H'
field theory", crucial to his theory of exchange in insula-
tors, is a theory of one-electron states for the
"non-magnetic lattice" which is supposed to have lattice
translational symmetry and consequently these states are
Bloch functions. We therefore consider this as a
one-electron band theory. Both Hubbard's and Anderson's
band theories are made in the Hartree-Fock approximation,
Hubbard's in the restricted (non-magnetic), Anderson's

in an unrestricted (magnetic) HF theory.

11-15 of the HFA as

In view of the known failures
discussed above, we certainly must answer the obvious

question, why even consider these HF band calculations

of Hubbard and Anderson? The answer is that the failures
result from actually applying the HFA in the way it was
derived, as a one-electron approximation, to deduce
physical quantities (specific heat, susceptibility, etc.);
whereas neither Hubbard nor Anderson use their HF
band-calculation in this way. Rather, they use their

one-electron operator in combination with important

 

two-particle terms (intra-atomic repulsion in Hubbard's

 

case, the same plus interatomic exchange in Anderson's
case) which, for narrow bands, actually dominate the
band-theoretic term. Hence our results of sections II
and III, namely that these HF approaches are unsatisfactory
within the context of the use made of them by their
authors, do not follow obviously (as far as we can see)
from previously known failures of the HF approximation.

It is convenient to discuss here the interesting work
of MattheissZI. At first sight he might appear to be

following the philosophy of Slater's group since he21

1/3

used Slater's p exchange (gnéspin-polarized) to calcu-

late the band structure of KNiF3. However, he didn't
follow that philosophy22 according to which one should
compare the predictions of their one-electron theory with
experiment and, if this results in failure, more refined
one-electron approximations should be attempted. Instead
Mattheiss correctly recognized the essential importance of

following Anderson and Hubbard in introducing the large

(intra-atomic) interaction terms along with his one-electron

term and considering the latter as a small perturbation,

when dealing with these very narrow band situation323.

l/3

However, Mattheiss' use of Slater's p approximation for
his one-electron terms is still arbitrary and unjustified,
and part of the motivation for our approach (section IV) is
to reduce, by theoretical means, the arbitrariness of this
potential.

The gist of our variational approach presented in
section IV is as follows. With Anderson and Hubbard, we
recognize that in our narrow-band situations certain
two-particle terms (including intra-atomic Coulomb inter-
actions) are much larger than the interatomic single-particle
terms essential to band structure, and cannot be approxi-
mated even crudely by effective single-particle terms for
most of the common physical observables. Hence we do not
consider applying a single-particle approximation to com-
pare with experiment. Rather, physical predictions are to
be made from a simplified "effective" Hamiltonian H which
includes certain types of two-electron terms. The
one-electron terms in H are called the band structure for
the interacting system, and are chosen variationally simul-
taneously with a similar choice for the two-electron terms.
It is in this simultaneous variational treatment of the
one- and two-electron terms that we differ from Hubbard and
Anderson, and it is this novel feature of our theory that
overcomes the difficulties in their theories pointed out
below, as shown in section V. Aspects of the present work

have been reported earlier24.

II. HUBBARD'S DERIVATION OF THE HUBBARD HAMILTONIAN; ITS

FAILURE.

The Hubbard Hamiltonian was derived5 in connection
with the problem of narrow-band electrons. Its properties
have been studied by a large number of authorszs. It
represents the simplest model of a crystal which contains
the tendencies toward itinerant behavior and toward
localization of the electrons. This Hamiltonian can be
written as

o + U E Nit NiI (2)

+
H = Z X b.. c. c
H ij 0 13 10 i

. . + .
where {1} refers to Sites; cio and ci0 respectively creates

and destroys an electron with spin 0 in the Wannier func-

. O O + O
O 1 . O = O 0
tion wl(£) at s te 1, N10 c10 c10 the corresponding number

operator; bij is a hopping or transfer integral; and finally

U is the intra-site Coulomb interaction (U>0). The

one-electron term causes the tendency toward itinerancy,

while the two-electron (interaction) term causes localization.
One can study the properties of (2) in two different

ways. In the first, one tries to calculate bij and U,

according to the actual derivation of the Hamiltonian.

The ”band theory" part of such a derivation is the calcula-

tion of bij (the one-electron part of Hubbard Hamiltonian).

9

10

In the second, one regards bij and U as parameters and
proceeds with the study of the properties of (2). It is in
this latter, phenomenological, sense that most researchers
have used the Hubbard Hamiltonian. In this sense (2) is
not restricted to narrow-band electrons; it can be an
approximation for any bandwidth, i.e., any bij/U’ including
bands that are wide compared to U. Our interest here is in
the first view, namely we are concerned with the principles
of the calculation of bij and U.

Let us outline briefly Hubbard's derivation of his

Hamiltonian presented in reference 5. For clarity we will

discuss this in the context of a single-s-band model.

The Hamiltonian H in this model i526
+ 1 +
H = X X h.. c. c. + — X E v.. c. c. ,c ,c (3)
ij 0 13 10 30 2 ijkl oo' ijki 10 30 lo ko

where hij is the matrix element of h - kinetic energy plus
the Coulomb interaction of the electron with the ions -

between wi(£) and wj(£); and

2
_ 3 3 * *
Vijk£ — j d r1 d r2 wi(£1)wj(£2) r12 wk(£1)w2(£2) . (4)

The Wannier functions wi(£) and the Bloch functions are

related by

b. .
”1(5) = N Z (exp-1§°§i)w&(g) (5)

11

where N8 is the number of sites, Bi is the position of site i,

and X goes over a Brillouin zone. Then the Hamiltonian (3)
k
can also be written as

l + +
H = X h a a — Z v a a .a ,a (6)
£.§° £0 2 §1§2E3£4 E10 R20 E40 E30
where
+ _ -g . _ +
a£0 - Ns E (exp+1k Bi) ci0 (7)
is the Bloch function creation operator, and
h = N'1 X h.. exp-ik-R . (8)
k s .. 13 — —ij '
_ -2
Vk k k k ‘ NS 2 Vijke eXp i(kl R +R2 Rj —3 Rkfi 'IR)
—l-—2—-3-—4
and
= R. - .
—ij -1 —3
If one makes the standard non-magnetic HFA to H, one
gets for the HF eigenvalues sip
HF _
SE — hE + XE , (10)
with
Ar :12. Vk' ”"3: .1: 12' 1:""12' r 1: Is." ' ”1’

12

where vk is the average occupation number for state k. Now,

with (10), eq. (6) can be written

a (12)

_ HF_ + 1
H — 2 (RE A 2

ango + — o'ak o'ak 0 °

a
o k _4 _3

X v a
k k k k 31 _2

)
R —1—2—3—4

As Hubbard5 noted, the presence of A in (12) avoids counting

k
the interactions of the electrons of the band twice, once
explicitly in the Hamiltonian and also implicitly through
EHF

k C

At this point it is convenient to write (12) in terms

of Wannier-function creation and destruction operators,

obtaining eq. (6) of reference 5:

H = R bij Cio Cjo + % R vijkll. Cio Cjo' Clo' cko
- Z (Zvijkg ' Vijzk) ij Cio Cko (13)

where

bij = Ngl E 8;? exp ikfgij (14)
and

v. = N“1 2 v exp ik-R. (15)

39. s .15 _l_(_ —-—32.

(so vjj = 1/2 for the half-filled band).

Having in mind that one is dealing with narrow-band
electrons, we can imagine a picture where the Wannier
functions are very much like the atomic s-functions and

these form an atomic shell with a radius small compared

13

to the interatomic separation. Then Hubbard argues that the

integral U : viiii

should be much greater than Vijki' if
i,j,k and 2 are not all equal. In other words, the
intrasite interaction U is much greater than all the inter-
site interactions. Neglecting then all intersite v's in the
last two sums, the Hamiltonian (13) becomes the Hubbard
Hamiltonian (2) plus constant terms, which are dropped.
The neglect of the difference between the Coulomb inter-
actions and their HFA (the second and third sum in (13),
respectively) is, for macroscopic systems, better than neg-
lecting the Coulomb interactions only, as we shall see.
However we shall now show that this approach has an unaccep-
table flaw.

The explicit expression for Hubbard's transfer integral
is

= h.. + Z (2 .) v

bij 13 k2 Vika ‘ Vikzj kg (16’

obtained from Fourier transforming (10).

Consider first the very simple case of two sites and
two electrons. In the usual situation where the bonding
state («(w1(r) + w2(r))) lies lowest, and the wi are real,

the expression (16) gives

_ 3
b ‘ R + 2 v1221

12 12 v1121 +

l
‘ 5 (l7)

V1212 '

All the quantities on the right side of (17) but the last

one v (inter-site Coulomb interaction) behave27 like

1212

14

exp -aR12 for large intersite separation R12 (a is a positive
constant). But the quantity v1212 goes like l/R12 for large
intersite separation. Hence the low-lying singlet-triplet
splitting 4bi2/U , calculated from the Hubbard Hamiltonian,
behaves as l/Ri2 for large R12 , in drastic disagreement
with the exponentially small behaviour of the singlet-triplet
splitting obtained directly from the exact Hamiltonian (3).
Explicit calculation of these splittings can be found in
Appendix Al.

For an abritrary number of sites NS , (16) can be
written

bij ’ hij + E vikjk ' Vijij vji

+ 2££ Vika sz Ezgjivikkj sz ' (16‘a)
For macroscopic systems NS+ m ; although X vikjk would
k
diverge (vikjk = R; as Rk+ 00) , these large-Rk terms are

cancelled or screened by the interaction in hij of an
electron with the nuclei, rendering the expression (16-a)
convergent. From (lG-a) the leading behavior for large

R.. = Igisgjl is seen to be bi' . At zero T,

—1 a : ’Vijijvji
at least for a variety of examples (e.g. a linear chain),
vji approaches zero as a power of Rij and oscillates sinu-
soidally with a wavelength 3 Fermi wavelength. So Hubbard's
transfer integral oscillates and vanishes as a power of the
intersite separation. This is again in serious disagreement

with the expectation that it should be exponentially small:

15

6,28,29

perturbation theory leads to a Heisenberg Hamiltonian

= - Z. J.. s.-s. (18)

H .
Heis ij 1] -1 —j

describing the low—lying states of H, where

J.. =

2

1111 Vijij) (19)

and

(Li)
t.. = h

1] ij + v11j1 + 1 Virjz ' (20)

The sum in (20) excludes the case £=i and j. Since

29

it can be shown in the narrow-band case that wi(£) is

30 31

exponentially small for I; - Ril+ w , (20) gives ti'

3

and therefore Jij being exponentially small for large RijBZ.

An even worse result would have been obtained if
Hubbard had merely neglected the interatomic part of the
Coulomb interaction (rather than the difference between it
and its HF approximation). Then the transfer integral bi'

3

would have been simply hij , which diverges for an infinite

crystal, fixed i and j , in accordance with the above

discussion.

III. ANDERSON'S LIGAND FIELD THEORY; ITS UNACCEPTABILITY

In this section we examine the concept of the ligand
field problem as discussed by Anderson and crucial to his
theory of exchange in insulatorsG. This problem was
defined6 as that of determining "the exact one-electron

states" for the crystal as a whole, excluding the effects

 

of exchange interactions between the magnetic ions. These
states are Bloch functions |k> and their energies 6k ; by
their definition6 they are to be "non-magnetic", i.;. each
wave function is to be a product of a spatial and a spin
function, with the spatial function and energy being inde-
pendent of spin. From these states one is to construct
"the exact Wannier functions" as in Eq. (5). Finally, one
is to carry out perturbation theory for the full crystal
many-electron problem considering interionic overlap to be
small. This procedure leads, according to Anderson, to the
low-lying energies of the system being predominantly the
eigenvalues of the Heisenberg Hamiltonian (18) where the

dominant contribution to the exchange "integral" Jij is

_ _ 2
J.. — 2|bij| /U + v. (21)

1) ljji

16

17

Here bij for i # j is the transfer integral or hopping inte-
gral as calculated from the ligand field Hamiltonian HSC:
bij a (wilnscle) , (22)

U is the change in energy needed, to zero order in overlap,
to transfer an electron from wi to wj . The spins of wi
and wj must be specified to define (22); Anderson did not
specify these - we will discuss the various possibilities
below. (We have simplified the notation to be appropriate
for a single magnetic band, which is sufficient for our
present purposes.) The first and second terms of (21) were
6

called "kinetic exchange" and "potential exchange",

respectively. The states |k>, 8k were defined6 (in the

"most formally exact" method) as eigenfunctions and energies

of the Hartree-Fock operator H in which the spins of all

HF
the magnetic ions are assumed parallel, i.e. in this most

exact method HSC = HHF and

HHFIk) = EEIR) . (23)

Anderson motivated this HF definition by saying that
certain desirable properties are expected to follow (for
example the non-magnetic electrons in the system, e.g.
associated with F- ions, can then be treated as core
electrons whose wavefunctions will not change very much
with magnetic excitations; again, the perturbation theory

is rapidly convergent when use is made of "the exact

l8

localized functions"). However these desirable properties
were not ghgwné to occur. In fact, it has never been clear
to us why a HF definition is at all appropriate: Since the
wi , which define the perturbation expansion, are to lead

to excited-state properties (the excitations within the
Heisenberg model, with energies per particle ranging over
.), and since a complete set of one-electron

3
states is necessary for perturbation theory (not just

an interval 3 J1

occupied states) the zero-T HF theory would not seem to

have variational significance33. The non-zero-T HFA (even in

the limit T+ 0), while variational, gives physically absurd

predictions for very narrow band systemslz'14’34. Neverthe-

less those absurdities were apparently not so obvious as to

prevent many workers from being misled.

29,35

Recently, it was shown that something is

definitely wrong with Anderson's definition. We present here

29

the simpler of these arguments. Consider a two-site model

having two electrons and two spatial orbitals W1 and w2,

27

like H in Slater's model where w1 and w2 are real ortho-

2
normal linear combinations of the ls-states centered at

the two sites. The zero-T limit of the HF operator with
occupied states wl...w£ is defined in general by its matrix

elements in an orthonormal basis set ¢1,¢2,... by

2' A
I¢1IHHF‘¢1"°¢1)I¢j’ (¢i|hl¢j) +k£l (¢iwklvl¢jwk) (24)

where

(¢wl3l¢'w') s (¢wlv|¢'w') - (¢wlvlw'¢') <25)

l9

and

(¢w|vl¢'w') Id3rld3r2 ¢*(rl)w*(rz)v(.r.12)¢'(51w(£2) (26)

The case of Anderson's definition (22) gives, from

(24)
o _
b12(+,+) : (inIHHF(wl+,w2+)Iwzo) (27)
h o = I
= 12 (28)
h12 + 2 v1121 o +

We used the symmetry and reality of W1 and w2 for the

hydrogen molecule which implies v also

1121 = V2212‘
h - * 3
12 — f wl(£) h w2(£) d r , (29)
where

_ 2 _ 2 _ _ 2 _ _ 2
h - p /2m e /I£ fill e /l£ R and v(r12)—e /r12 .

2"

The first difficulty presented by the result (28) is its
strong dependence on the spin of the states between which

the matrix element is taken: h12 and v are of the

1121
same order (first) in the overlap between the atomic

functions. Furthermore, the exact exchange integral J12

as shown in Appendix A1, is approximately

J (30)

—. _ 2 .—
12 ' v1221 ZtlZ/(Vllll V1212)

20

for small overlap, where

t = h (31)

12 12 + v1121 '

Expression (30) is like (21), but with b12 replaced by
(31); thus it is different from either of the values
given by Anderson's scheme36. Clearly then his theory
is not defined precisely enough for his purposes, and
neither of the choices (28) is correct within Anderson's
picture, the error made (which is first order in the
overlap) being of the same order as terms retained.
Although unsatisfactory, we note that Anderson's result
is far superior to Hubbard's. Fuchikami35 came to a
similar conclusion about the unsatisfactory nature of
Anderson's definition37.

Consider now the HF operator obtained by occupying
antiparallel spin states. The corresponding transfer
integral is

0' ._
b12II'I) ‘ (wlOIHHF(w1+’W2+)Iw20)

(32)

independent of the spin 0 . Hence the hopping integral

calculated using the HF operator with occupied Wannier

21

functions having antiparallel spins leads to correct
exchange splitting via (21). Fuchikami3S apparently
noticed the analogous point for her similar, but
slightly more complicated two-site model.

However, contrary to Fuchikami35, one cannot con-
clude from these special models that the HF operator
for the antiferromagnetic state is correct in general
(one can conclude only that using parallel spins is
not always correct). In fact, in certain cases (e.g. MnO)
it is impossible to have all the nearest-neighbor pairs
be antiparallel. Thus it would be impossible in those
cases to satisfy Anderson's requirement6 that the exact
one-electron states be solutions of a Schr6dinger equation
for the crystal as a whole simultaneously with this
antiparallelism of all near-neighbor pairs.

Furthermore, in any such HF scheme, the states do not
satisfy, even approximately, Anderson's requirement of
spin-independence. We can see this directly in terms of
our simple two-site model. Consider the parallel-spin

case. The HF equations are

(wv|HHF(wl,wz)|wu) = av av“ (33)

so if the eigenfunctions are written

w = Z A? w. , (34)

22

the coefficients Aio are to satisfy

Z'(w10 IH HHF(W1'w2 )Iw. JO,) Ajo . = e A. . (35)
jo

It is easily verified from (24) that the matrix elements

7 ° = I
(wioIHHF(wi+’w2+)ijo ) are zero 1f 0 o and that

h o = +

| 11 + V1212 ' v1221 '
HF W1+'w2+) w1= (36)
R11 + v1111 + v1212 ' 0 “ I'

io|H

thus with (28) we have all the matrix elements of HHFIW1+’w2+)
in this model. The eigenstates of HHF(w1+,w2+) (1n

our limited model space) are

_ -% ik
W — 2 (W1 + e w2) 0

k0 I k = OI”T I (37)

0'

(as easily seen from the symmetry), with energies, from (28)

and (31):

ik
11 + v1212 ' v1221 + e h12 ° = I
E:kcj = ik (38)

R11 + v1111 + v1212 + 9 (R12 + 2 V1121) ° =

h

Finally we see that (37), (38) provide a solution to the

23

HF equations (33) with occupied states

W = W , W = W (39)
because from (24) and (37)
(¢i|HHF(wl+’w2+)|¢j) = (¢iIHHF(w0+'Wn+)I¢j) ' (4O)

(HHF(wl...w2) is invariant under a unitary transformation
of the occupied orbitals, as is well-known). Thus

ik

+ 2 e v1121 - (41)

€k+ ' €k+ = V1111 + v1221
Since (41) involves the intra-atomic Coulomb integral
Vllll' the difference between up and down energies is very
large, even for small overlap. Clearly the intra-atomic
nature of this term implies that it will occur independently
of the relative orientation of the occupied spins. Pic-

torially, the viiii

occurs in ck+ because wi+ is unoccupied;
as is well-known (ref. 8, secs. 5-2, 6-2) the HF equation
for an unoccupied state corresponds intuitively to the
Schrodinger equation for an ddddd electron. One might

note that the wave-functions do not show this magnetic
property in this model: the up and down wave-functions

at site 1 are wlo = w1(£)aO (the same spatial function

for both spins). However, it is clear from the above

24

discussion that this non-magnetic property of wave-functions
results only from the restricted nature of the basis set
used to define our model; if for example one enlarged the
basis set to include 25 states wi,wé (giving 4 spatial
orbitals), then the HF wave-functions wig will be of the
form wig = w:o(£)ao where v = 1,2 (corresponding to ls and
23), k = O,n again, but w:0(£) will be expected to depend

on 0 even for zero overlap38.

Finally we should address ourselves to the apparent
conflict between the conclusions just reached about the
unsatisfactory nature of Anderson's theory, and statements
recently made supporting it35’39'40. Of these works we
are principally concerned with those which consider the
crystal as a whole. In fact there is no conflict with the

35 O I
, because a) her Wannier functions were

work of Fuchikami
determined using experimental output rather than by
Anderson's HF method and b)she considered only 3d cation
bands (without higher ones). Hence she did not deal with the
aspect of Anderson's method that we have criticized. fkfl?
avoidance of this criticism is of course not without cost:

shereplaced a fundamental theoretical question, that of

determining theoretically and therefore predicting the

 

best Wannier functions,by a phenomenoloqical procedure.
We also note that a glaring defect in Fuchikami's calcula-
tion is the use of a value of U appreciably and apparently

arbitrarily reduced (in quantitative terms) from the

25

carefully calculated value UO . (The value of the exchange
integral is very sensitive to this change since the

kinetic exchange and potential exchange terms very

nearly cancel if the U0 value is used.) The physical
effect invoked as the cause of this reduction is real

35

(it's called "electron rearrangement" by Fuchikami and

was referred to as a "correlation effect" by Hubbard

et al.41). Such an effect can be included in a perturba-
tion theory provided higher bands (e.g. 4s cation bands)

are included in the basis set - in such a case the magnetic
properties of the Wannier functions would show up in
Fuchikami's HF method, as discussed earlier in this section.

40

Gondaira and Tanabe seem to have followed Anderson's

HF method; unfortunately they did not define their HF
operator (in that the occupied spins were not defined) and
therefore their consideration cannot be considered as a
valid investigation of Anderson's approach. Finally the

treatment of Anderson's approach by Huang Liu and Orbach39

.35

followed that of Fuchikami and so is subject to the

deficiencies of that method.

IV . VARI AT IONAL APP ROACH

Our initial approach consists of finding the varia-

tionally best Hamiltonian of the form

~ ~ ~

~ ~+
H “ X bvi,vj Cvio Cvjo + Z Uv Nvi+ Nvi+ (42’

~

where v is a band index, i,j label atomic sites, Cvio

creates an electron in the Wannier function wvi , with
~ ~+

spin a , N . = c 5 and b

. . U ar varia-
V10 v10 V10 ’ v e the

vi,vj
tional parameters. Further, wvi are taken as orthonormal
linear combinations of a basis set of one-electron func-
tions (which can be chosen arbitrarily), the coefficients
being also variational parameters. In other words, we
take the trial Hamiltonian essentially of the form finally
assumed by Mattheiss21 as a sum of a band structure
Hamiltonian, namely the one-electron operator H(1) in (42),
plus a Hubbard-type of interaction term (involving the Uv)°
But instead of choosing the band structure Hamiltonian
arbitrarily, and in a way conceptually associated with
nearly-free electrons, we choose it variationally, treating
it on the same footing as the U-terms.

To accomplish this we use the so-called variational

principle of statistical mechanicsg, namely, that the trial

26

27

free energy at temperature T and chemical potential u satis-

fies the inequality

“1 > FIH] (43)

tr 5(H-H) - B

F[H] in tr exp-8(H - uN)
with

‘ exp-8(H - UN)/tr exp-8(H - UN) (44)

'0
II

for all Hermitean H. Here H is the Hamiltonian taken as
exact and which is being approximated by H. By this
principle, the best estimate of the free energy is the
minimum of F[H], and this is the criterion for determining
the best parameters in H.

Clearly the resulting one-electron Hamiltonian ~(1),
namely the one-electron part of the "best" (42) is by
definition variational, it is a property of the crystal as
a whole and is non-magnetic. Further we will see that the

"best" (42) gives the correct behavior for single narrow-

band models provided the exact kinetic exchange,

_ 2 . .
K° ' : "2tij/(V- " ) I (1%)) (45)

13 1111 Vijij

dominates the potential exchange Vijji' Therefore for such
~(1)

models the (one-electron) eigenstates of the best H

could provide a definition of the ligand field states which

represents a major improvement over the HF definition6'35.

However the limitation to cases with dominant Kij is unsatis-

factory on general grounds (in fact there are cases where

28

39). Further for more than

this is expected to be violated
one narrow band (e.g. a "degenerate" band such as a 3d-band
in NiO), the trial Hamiltonian (42) will be unsatisfactory
on the additional ground that Hund's rule, essential to
magnetism in insulators, would be violated43.

These difficulties should be overcome by adding
potential exchange type terms to (42); also generalizing

the one-electron terms (with no additional serious

increase in difficulty) we obtain as our trial Hamiltonian

~+ ~ ~ ~
N

H = Z bvi,uj Cvio cujo + z Uv Nvi+ vi+

_ _ U . . c , _ . .
where U . . are additional variational parameters which

VlIUJ
physically appear as effective potential exchange matrix

elements. It should be realized that the interaction
terms in (46) are of a much simpler form than the (exact)
Coulomb interactions.

To find the best H we must solve the stationarity

equations (3/3l)F[H] = 0 which may be written42

‘0?

3

tr 3

(H - H) = o (47)

>’

with fixed temperature T and chemical potential u ; here
1 stands for any one of the variational parameters. The

derivative of the density matrix 5 , (44), with respect

29

to A is of the fOrm

3d _ 1 _ ” __ -B(H - uN)
3A ’ z (1 0 tr) 3A 3
where
2 = tr e-B(H - UN) (49)
and
~ -B(H + 51 §§-- uN) ~
3 -B(H - uN)_ 1. 1 3A -B(H - uN) (50)
57 e - 1m. Ei'le -e ],
6A+O
. . 44
Wlth the use of the expan81on
A+B _ A 1 1 x1
e — e [1+]o dx1 B(xl) + [o dxl [o dx2 B(xl) B(x2)+...] (51)

(A and B are operators, and B(x) E e-XABe+XA) , the

expression (48) becomes

8
3

'02

= -s (1 - 5 tr) 0 I; dx QA(x) (52)

>2

where QA(x) is defined as

91‘“) E exB(H - uN) %%_e-XB(H - nu) . (53)

Then it follows that the stationarity equation (47)
becomes
~ 3H =
A

I: dx <Qx(x) (H - H)> - <H - H><§—> o . (54)

30

We have used the following notation for the thermal average

of an operator A:

<A> S tr

02
w

(55)

The fact that H and %% do not commute with each
other in general, makes (54) much more complicated to
calculate than if they did.

We have so far investigated this approach only for
the single-band problem, taken for simplicity of discussion
to be an s-band. We further have limited ourselves to the
narrow-band or small-overlap region. Our investigations
are described in subsequent sections.

It is worthwhile mentioning that recently it has been

shown42

that a large class of approximations in statistical
thermodynamics that are based on (43) yield expressions

for the macroscopic quantities of the system that are con-
sistent from both the statistical mechanical and thermo-
dynamical points of view contrary to previous claims. A
question in this regard arises because the variationally
best parameters that appear in H are generally T and u
dependent, unlike an exact microsc0pic Hamiltonian. That

result was important in the considerations of the present

paper.(1t answers questions like, should the approximate

entropy be calculated as -%T F[H(b)] or -ktr B(b) ln B(b) ,
guaranteeing that they are the same. By H(b) and 5(b) we

mean the best H and the respective density matrix according

to (43)).

V. EXPLORATION OF THE VARIATIONAL APPROACH WITHIN THE

SINGLE-BAND PROBLEM

The exact Hamiltonian for this problem is given by

(3); and according to our approach the trial Hamiltonian

should be:
~ ~+ ~ ~ ~
H = 2 b.. c. c. + U 2 u. N.
i,jo 13 10 30 i 1+ 1+

ij 00' 13
Although the Bloch functions wk(£) are physically
invariant for a single band (aside from a phase factor

exp i Yk) , the Wannier functions wi(£) which are to be

constructed as

g g exp i(—_¢§i+yk) wk(£) (56-a)

wi(;_)=NS
are not invariant because of Yk . Then, in principle one
should ask for the best wi in the sense of (43), leading
to an enormous number of variational parameters. However,
it seems natural to demand the Wannier functions to have,

in common with the atomic ls functions ai , the properties

31

32

of reality and invariance under inversion through 3i ;
and also to go continuously to ai as the interatomic
separation goes to infinity. Then one proves (Appendix
A2) that for the single band case Yk is constant, leaving
no physical arbitrariness in the choice of the Wannier
functions, the variational parameters being only bij' U
and Uij . (We can then drop the tildes on the right side
of (56).

Naturally the best parameters will be found to be
functions of the various integrals in the exact Hamiltonian
and of B and u . In particular, the leading terms of the
expression for bii and U are expected to be zero order in
the overlap; similarly bij and Uij (for i # j) are expected
to be mainly of the first and second order in the overlap,

respectively. One can see that this is very plausible:

Suppose the exact Hamiltonian contains only the terms

X h N + V 2 N N + X. h E c+ c - (1/2) Z'v
i 11 1 i 1+ 1+ ij 1) o 1o 30 ij 1331

+ + _ . . 9
ga?iocio'cjo'cjo , where V _ Viiii . Then the m1n1mum
of the trial free energy is attained
for bii = hii’ U = V, hij = bij and Uij = Vijji, in which
case min F[H] = F[H] . The quantities hii and V are
clearly zero order in the overlap, and hij'vijji for 1 # j

are first and second order in the overlap45. So that in

the case of the full Hamiltonian (3), the parameters bij
and Uij (i i j) should be zero in the limit where the

overlap goes to zero. In accordance with this expectation,

33

we will assume in the remainder of this paper that bij

and Uij+ 0 as the nearest overlap A+ O ; more precisely,

we shall assume that bij = 0(1) and Uij 0(A2) . The
latter assumption will be shown to be a consistent one in

each of the cases studied.

A. Zero-Bandwidth Hamiltonian; All Temperatures

 

 

To get some insight into more complicated cases we

studied first the simple case in which we keep only terms

13 in the exact Hamiltonian H

to zeroeth order in overlap
(3); in other words, we consider the atomic limit or

zero-bandwidth exact Hamiltonian:

_ l
Ho — E [buni + v NHN1+ + ngi vijij NiNj] (57)

where N1 = Ni+ + Ni+ , h11 = hii , and V E v. In

iiii
accordance with the discussion of the previous paragraph,

the trial Hamiltonian is taken of the form

H = E [biiNi + UNi+N

0 Z Hi ' (58)
1

1+]

This is also the atomic limit of the Hubbard Hamiltonian.

The consistency of the assumption bij = Uij = O , i i j ,

will be seen in secs. B and C below.

34

Because (58) is function of number operators only,

 

 

8H
572 commutes with HO and (54) becomes:
3H0 ~ . 3H0
<31 (HO-Ho)> — <Ho-HO><3A > = 0 (59)

In particular for A = bll , making the restriction that

bii is independent of i,as is hii by the lattice transla-

tional symmetry,

(v-U) é (<NiNj+Nj+> — <Ni><Nj+Nj+>)

+ (h ) Z (<NiNj> — <Ni><Nj>)

11'b11
J

1 ' _ .
+ 5 £2 vj£j£(<NiNjN2> <Ni><NjN£>) — 0 , (50)

similarly for A = U
(V'U) g (<Ni+Ni+Nj+Nj+> ' <Ni+Ni+><Nj+Nj+>)

+ (“11'b11) g (<Ni+Ni+Nj> ' <Ni+Ni+><Nj>)

1 ' _ =
+ E E2 Vj£j£(<Ni+Ni+NjN£> <Ni+Ni+><Nj ”2” 0 ° (61)

The traces in (60) and (61), according to (44)and
(55) are to be taken over the eigenstates of Ho (which are

also eigenstates of H0 - uN). As pointed out by Kaplanlz,

35

a complete set of eigenstates of this operator is given by

a set of Slater determinants obtained by occupying Wannier
functions in all possible ways. Since Ho - uN = Z (Hi - uNi) ,
the density matrix 1

-B(fi -uN) -B(H -uN)
6 = e o / tr e o (62)

5 =15. (63)
1.

where

-B(H.-uN.) -B(H -uN )
51 = e 1 l / tr e 1 l - (64)
Using (63) it is easy to formally evaluate (Appendix A3)
the thermal averages in (60) and (61). Finally we obtain

the following:
U = V (65)
(66)

where

-B(b11-u) -B(2b11-2u+V) -B(b11-u) -B(2bll-2u+V)

fi=2(e + e )/(l+2 e + e )

(67)

36

is the average number of electrons per site in the trial
Hamiltonian model. Equation (66) is, in fact, a self-
consistent equation for b11 , given the chemical potential
and temperature. In particular there exists a special

value of u namely “0 for a given T such that fi = l . This
is of special interest because it is the zero-bandwidth
limit of the half-filled band case to be discussed later in
this paper. So in this case we have found, for all tempera-
tures

b = h11 + Z . (68)

V . .
ll j¢l 1313

From (67) and (68) we infer that the special value no of

the chemical potential should be

I:

ll
:23"
4.

MI (I

+ 2 v . . (69)
. l
3%1 13 3

independent of T.

Kaplan and Argyres, in appendix B of reference 13,
showed that the ground state of the Hamiltonian (57) with
the number of electrons equal to the number of sites, has
one electron on each site, and the many—body ground state
energy Eg(NS) is
1 I
Eg(NS) - Nsh + 7 E

11 .j Vijij

The ground state energies of the system described by (57)

37

when we add one electron to or remove one from the NS

electrons already there, are

l I
E (N +1) = (N +1)h + —»2 v.... + Z v + v
g s s 11 2i,j 131] k#1 lklk
12' z
E(N-1)=(N-l)h +- v....— v
g s s 11 2 ij 1313 kfl lklk

Then the chemical potential u at zero T for HQ with NS

electrons is
u = its (N +1) - E (N -1)]
2 g s g s

= h + §.+ Z (70)

1 . v .1. .
1 J#1 13 3
This result (70) although derived for zero-T should be
valid for all temperatures. In fact we have derived
through the expression

-B(Ho-uN)
tr e (N - NS) = 0 (71)

the chemical potential for two sites and rings of three,

four and five sites, for any temperature:

_ v
two u — h11 + 7 + v12
th u = h + !.+ 2v

ree 11 2 12
f u = h + 21+ 2v + v
°ur 11 2 12 13

38

+ 2v .

+ 2v 13

five u = h11 +

hfl<

12

Here v12 and vl3 are the first and second neighbor inter—

actions vijij ; the result (70) is obviously a generalization
of these.

Expressions (69) and (70) plus the argument of the
previous paragraph suggest strongly that both Hamiltonians,
the exact Ho and the best Ho (which we will call Ho(b)) are
exactly half-filled band Hamiltonians.

From appendix A3 we see that the best value of the
parameter U is found uniquely for any density of electrons
fi, except zero. The fact that the effective intrasite
interaction U is strictly equal to the exact intrasite inter-

action V is reasonable for the grand

canonical calculation. Our trial Hamiltonian Ho does not

 

contain any intersite effective interaction; hence one elec-
tron can, thermally or by other means, move to an excited
state by going to 32X other previously singly-occupied site,
with the same probability. And because of this lack of
correlation between doubly-occupied and empty sites in Ho,
we would expect that the interatomic interactions in Ho

would enter our single interaction parameter U, in a canonical

 

calculation (for fixed N = NS , in our case), only through

an average

__l_2v
N — 1 j ljlj '

39

which goes to zero as NS goes to infinity. (Vijij+

as Rij+ w .) This expectation was found to be born out

in straight-forward canonical calculations, for two sites
and rings of four and six sites. The results are as

follows:

two U = V - v12
four U = V - l (2v + v )
3 12 13
“i U = v — i (2v + 2v + V )- (72)
° X 5 12 13 14

This also checks the fact that the canonical ensemble
approaches the grand-canonical ensemble when the number
NS increases to infinity.

When one studies the properties of the (phenomenological)
Hubbard Hamiltonian, one in general neglects the term

b11 : bii , because there it only represents a shift in

the chemical potential; that is, b11 becomes an unimportant
constant. Here it is different; because of the presence

of the difference (H - H) in the variational principle (43),
bll does not appear only in the combination b11 -u , there
being terms containing b11 alone. We have seen already

that, in fact, the best value of bll given by (70) has been

chosen by the variational principle to make Héb) have the

same number of electrons as HO . Its importance then has

40

already been shown. Another even more important reason

42

to have b11 in our scheme is related to the consistency

between statistical mechanical and thermodynamical expres-
sions for the macroscopic quantities of the system such as
internal energy, entropy and average number of particles.
It has been shown (in reference 42) that the presence of
bllN terms in the trial Hamiltonian H is necessary and

sufficient to get the above consistency, b being a

ll

variational parameter.

(b)

The (approximate) entrOpy S correspondent to Ho(b)

can then be either calculated from F[Ho(b)] or from the
(approximate) density matrix E(b):
3“” = — g3.- F[H0(b)] = -k tr 5“” 2n 5””,(73)

The specific heat follows immediately, and we find it is
independent of the Coulomb intersite interactions ijkj ;
it is the same as calculated from a Hamiltonian which con-

sists purely of intrasite interactions V:

(b) -12 .812
CV /kNS — ( 4/cosh 4) . (74)
To evaluate the susceptibility, we have first to add
the magnetic field term, namely -h§Mi = —h:ZL(Ni+ - Ni+)/2’
to both the exact Hamiltonian and trial Hamiltonian, and

similarly, find the best parameters, as function of terms

41

of HO, temperature, chemical potential and the magnetic
field (h is equal to guB times the magnetic field). In
the half-filled band limit they turned out to be the
same as those of (69) and (70), and consequently inde-

pendent of the magnetic field. Again42

8h 0 — M (75)

_ h . . .
where M — f g (Ni+ Ni+) IS the magnetization. The

zero-field susceptibility (per site) is

8V

x(b) (e 4/cosh §%) , (75)

Nun

and is also independent of the intersite interactions.

It would be interesting to compare thermodynamical
properties of this Héb)with those of HO for an arbitrary
number of sites. Since one cannot calculate the exact
free energy correspondent to the exact Hamiltonian Ho ,
in general, this cannot be accomplished. Recently,
thermodynamical properties of a linear chain Hamiltonian
consisting of intrasite and nearest neighbor Coulomb
interactions, V and v12 , respectively, have been calcu-
lated by Tu and Kaplan46. They plot the specific heat
CV/kN and the inverse zero-field susceptibility

(XV/NU; )-1

n = Vlz/V . The ones for n = 0 correspond exactly to

as function of kT/V, for different ratios of

42

our results here, and should be compared with the others
for different n's . The susceptibility is practically

insensitive to n when 0:n<% ; the specific heat is also

quite insensitive to change of n in the same interval

although slightly more sensitive especially close to

n = % . This value n =

‘5
Bari47, because for n > % the ground state of this
V

l-'

is a critical one as noted by

Hamiltonian (containing and v interactions only)

12
consists of alternating empty and double occupied sites.
Thus for n > % the variational approximation Héb) is
extremely poor since the approximate ground state con-
tains singly-occupied sites (U = V>0) ; however, as we

have said, the variational approximation is quite good

for rather large n(2%) .

B. Narrow Half-Filled Band: The "Best" Hubbard

 

 

Hamiltonian

 

In the narrow band region the overlap between the
atomic functions is very small and all the terms in
H, (3), which depend on the overlap are small compared
to the others. For convenience we separate the exact

Hamiltonian H in two parts:
H = H + H' (77)
o

where H0 is the zero overlap part given by (57)and H' is

I
defined by (2 means a sum with all indices different)

43

H' = Z X [h.. + v....(N.- + N.-) + X v. . N 1c? c.
0 ij 1] 1131 10 30 k#i,j 1k3k k 10 30
' + +

+ f g Ej Vijjilcio Cjo + °j6 Cio)cio Cjo ’ Nio Njo]

+ i X Z. v .. (cT- c - + c+- c -)c+ c (78)
2 o ijk i13k 10 k0 k0 jo io jo
1 1 + +

+ 5 go. Ejkl Vijkl Cio cjo' Cfio' Cko '

We begin our exploration of the narrow-band case with
a trial Hamiltonian which is of the form of the Hubbard

Hamiltonian (2). In other words we take the Hamiltonian

(56) w1th Uij = 0 , and call 1t HH
~ I +
HH “ Z (bllNi + U Ni+Ni+) + Z Z. bij Cio cjo (79)
1 o 13
which for convenience is written as
H=HO+T (80)
where H0 is (58) and T is
~ I +
TEX): b..c. c. . (81)
0 ij 1] 1o 30

As we argued before, the best bij(i#j) are expected to

be proportional to the overlap in the narrow band limit, so

that when H'+ 0, T+ 0 too. We can then say that T is

"small" compared to H0 and expansion techniques can be

used to solve the stationarity equations (54).

44

We propose to solve these stationarity equations for
A=b11, U and bij(i#j) in two limiting cases: 1) high
temperatures, in the sense that kT>>Ibij| for all i and j;
2) low temperatures, in the sense that kT<< nearest

neighbor lb. I . In both cases |b..|<< U .
1 13

j
B.l High—Temperature Limit

It is convenient to define

 

Ho 5 Ho - uN (82)
~ 88 -8(8 +T)
and R E e 0 e o
-B(HH-UN) -880 ~
such that e = e R .
Consequently
i = 2 <fi>
~ 1 ~ ~
0 = i p R (86)
<R> 0
o
where
~ ~8fio
Zo = tr e (87)
~ -BHO ~
00 = e /Z0 (88)
and <...> 5 tr 5 (...) . (89)

45

With all these definitions the stationarity equations are

written as

~ 1 ~ ~
<R>o f0 dx<RQA(x) (H HH)>0 (90)
~ ~ ., afiH .
- <R(H ‘ HH) >O<R fi—DO: 0 .

According to (89) the thermal averages in (90) are to be

evaluated with respect to the density matrix po s exp-BRo .

We should point out that the equations (90) are still exact.
In this high temperature limit we expand

exp i x B(RO+T) in powers of Blb..I by using Eq. (51) and

13
keeping only terms of first order in Blbijl:
:x8(fio+T) ixBfio x ~ _
e = e [1 1 I0 ds T(+s)] (91)
where
~ 588 ~ ~3880
T(s) E e T e
I SBU(N " N ")
k0 £0 +
= Z X b e c c . (92)
0 k2 k2 kg 20
We refer the reader to Appendix A4 for the proof of the
last equality in (92).
Defining
sBU(N - - N -)
: k0 20
gk£6(s) — e (93)

46

to simplify notation, we have

18(fi +1) iBR '
- +
e 0 = e °[1 : B X Z bk2 1: ds gk£5(+s) ckO C20] .(94)
o kl
Similarly R becomes
~ 1 ~ ‘ . l +
R = 1-8 [0 ds T(s) = 1 -B 2 Z bk2 [0 ds gk£5(s) ck0 c1OF .(95)
0 k2
Immediately we see <R5 = 1 because
<c+ c > = o , for k ¢ n . (96)

ko no 0

With (94), QA(X) defined in (53) becomes

' x +
QA(X)=AA(X)+B g X bkz Io d3 [9k15(’5) 9k25(x) cko C20 AA(X)

k2
-A() -()+ 1 (97)
A x gkzo S cko C20
where
x88 ~ -x88
AA(x) s e ° gE-e ° . (98)

In accordance with previous discussion we assume the
best parameters bij(i # j) to be proportional to the overlap
between atomic functions i and j; we then calculate (90)
neglecting terms which are higher than first order in the
overlap. In so doing we easily find that for A = b11 and
A = U the stationarity equations (90) become identical to

(60) and (61) respectively; we already have the solutions

for those equations:

47

U = v (99)
b = h + E v . . . (100)
1 .
ll 1 3#1 1313

This last one holds only in the half-filled band regime.

For A = bij(i # j) eq. (90) becomes

1
+ r
[0 dx g <g —(x) c. c. (a - HH)>O +

ijo 1o 30
1 1 x + +
+ Bag. £2 bkz Io d“ {Io dS< [9k26'('5) 9k25"x’ cko'clo'gij3(x)ciocjo
- g -(x) c+ c g - (s) c+ c ](H - H )> - (101)
ijo io jo kzo' ko' 10' H o

l
+ + ~
-Io ds<gk£5,(s) Cko' cko' gij5(x) Cio Cjo (H _ HH)>o +

+

+ ~ —
+ (qua-(X) cko' Czo' cio cjo>o ‘3 " “3%) ' 0 -

we then proceed as in Appendix A5 to solve (101) for bi';

 

3
we find
b = [1 - :iiii f(BV)l-l t (102)
ij V ij
where tij is given in (20), and f(BV) is
-8V
1 - (1 + BV)e
f(BV) = _ _ ; (103)
1 + 8v e 88v _e 8v

48

f(8V)+ 1(0) as 8V+ 00(0). We see that Eq. (102) is con-
sistent with our assumption bij = 0(A). Then, particularly
if kT<<V the expression for bij will reduce to

V. .. . _l
= — 1 l .
bij (1 -—€%¥1) ti] . (104)

The fact that the best U turned out to be equal to V
is again very reasonable, because kT>>Ibij| for any pair
1 and j means that the electron can go from site to any
other singly-occupied site with equal probability; then the
same discussion of section V.A on this matter applies here.
The value of bll is also understandable: as can be shown,
” (b)

it again makes both Hamiltonians H and HH

have the same number of electrons to first order in bij

(the best HH)

(in the half-filled band case).

B.2 Low-Temperature Limit

 

In order to calculate the quantities appearing in the
stationarity equations (54), we must first discuss the

eigenstates of H When the number of electrons N is

H’
equal to the number of sites NS , the ground states of
HQ with U>0 have one electron on each site, and no
doubly-occupied or empty sites, the degeneracy being 2N.
Adding small bij terms to it will remove the degeneracy;

6,29,48 (for

second order degenerate perturbation theory
the energY) shows the resulting energy levels are the same
as those of a Heisenberg Hamiltonian (except for a constant)

with the exchange integral aij = -2b§j/U .

49

One can divide the complete set of eigenfunctions of H0
in two sub-sets: G - the ground states; and E — the excited
states. Consider the following:

o sbll

:1
v
m
(I
z

fiH|fi> = Enln> (105)

where Ino> is in G , and En and In> are the eigenvalues

and eigenfunctions of H , resulting from the removal of

H
the degeneracy of the ground state of HO . These contain
the low-lying eigenstates.

If one defines idempotent operators P and Q such that

P? s G (106)
and

Q? s E
for all wavefunctions W so P + Q = 1, then one can write
the eigenfunctions In> as 29

|fi> = E [1 - Q :_1___ Q $1 P B > , (107)
n ~ 0
HH-En

where Cn is the normalization constant, which can be con-

sidered real. One then can make the following expansion29
” ~ ~ k
Q:—1:0 = ozl—zo Z (-)k[Q:—l:o (T - AEn)Q] . (108)

HH-En Ho-eo k=0 HO-eo

50

~

where AEn = in - so . By keeping only the first term

(k=0) in the series (108), one obtains the approximate

eigenfunctions of H in formal second order degenerate

H I

perturbation theory:29

In>= c (1-0.,1~
n

H -E

00

 

0%] Plfio> . (109)

In the class of the operators Q , one can distinguish

(V) ’ with v = 1, 2, 3, ... which project

th

the operators Q

functions onto the v excited states of the unperturbed

Hamiltonian HO , the energy of these states being

Nsb11 + vU . Since our perturbation term T , (81) , involves

a hopping cf cjO from site j to another 1 , it is clear

10
that the operator Q in (109) is in fact a Q(1)

operator.
For clarity, it is convenient to rewrite (109) with this

new information:

 

|fi> = E [1 - 0(1) 1 0‘1) &] Plfio> . (110)

n ~ ~
-e
Ho 0
The orthonormalization condition <m|n> = 6 gives

nm

b2
~ ~ ~ ~ ~ _ ' 00 00-
c2 = [1 + 1— <n |PTQ<1)TPIn >1 1 = [1 + Z —£1.A13] 1 (111)
n U2 0 o i' 2

with

2
b. O . .
<mOIPTQ TPIno> — X —61 An 5

ij

nm

\

51

where
13 _ ~ + + ~
An _ go' <no| ci0 cjO Cjo' Cio' no> . (113)
*
Here we have considered b.. = b.. = b.. .
1] 31 13 2

b. O O 0
~ '
It is clear from (112) that In0> and X _%1 A33
13
are, respectively, eigenfunctions and eigenvalues of the

~

operator Heff

~

_ 1 ~ (1)~
Heff — U PTQ TP (114)

which, in Appendix A6 is shown to be identical to the
Heisenberg Hamiltonian, with Jij = ~2bij/U , except for
constant terms.
It follows from (105) and (110) that
2

~ b.. ..
E = - 2' —$1 A13 + N b
U n s

n .. (115)
1]

ll '
These are then equal to Heisenberg Hamiltonian eigenvalues,
except for a constant. (By constant here we mean inde-
pendent of Ifio>) . To simplify the calculation we are

about to make, let us consider only nearest neighbor hopping

bij=b12 . Then (115) becomes
~ biz
En = “‘6‘ An + Ns b11 ' (116)
where
= ij
An 2 Z An , (117)

<ij>

52

<ij> denoting sum over nearest-neighbor pairs.

For an arbitrary number of sites one does not know how
to calculate the eigenvalues and eigenfunctions of the
Heisenberg Hamiltonian, which means, in our language here,
that we don't know how to evaluate An and (30> for an
arbitrary NS . Nevertheless as we will show, our calcu-
lation of the best parameters can be accomplished without
knowing explicitly (50> and An .

Since we are interested here in temperatures which are
low compared to U and lblzl , we perform the traces in
the stationarity equations (54), only over the low-lying

eigenstates of H We then get

H O
biz ~ 2
B_U— (An+A£) ~ BHH ~ b12 ~ ~
e {<nI5-X— In> [T (An " A2,) - <2|H|£>l
n,£
1 ~ ~
+ Io dx<nIQA(x) HIn>} = 0 . (118)

Because of the restriction on the traces, bllN appears only
as a constant term (=b11Ns): thus bll cannot be determined
variationally within the present approximation; consequently
we drop it out of the calculation. (But see section B.3).

The last term in (118), containing the integral is

found to be

 

1 Bi (A —A ) ~
~ ~ _ U n m _ ~ 3H ~ ~ ~
Io dx <nIQA(x)HIn> — z e 1 <n|-—fl |m><m|H|n>
m 2 3A
E12 (A _A ,
U n m
~ b2 ~ BHH ~ ~ ~
+ kT X (E — 12 A ) (“'51- |¢><¢|H|n> , (119)

53

where |fi> and |$> are eigenstates of H which originate

H
from G and E , respectively. Then we proceed and calcu-

late the several matrix elements in (118) and (119). We
started with the matrix element of the exact Hamiltonian H

between the eigenfunctions of the trial Hamiltonian HH.

In accordance with our previous discussion we assume that

b12 is 0(A), and since we are doing this perturbation theory

which is second order in T , we neglect terms in this matrix

element which are of higher order than A2 .

Dropping the P operators which appear in (110), because
P|no> = Ifio> , and defining, for nearest neighbor i and j
only

12 ‘ Vijij

, t z t.. and X (120)

V 12 ‘ 13 12 ’ Vijji

we have, using (110)49 ,

~ ~ ~ ~ ~ ... 2 ~ (1 ~ ~
<m|HIn> = Cncm [<moIHIno> - fi'<moIHQ )Tlno> +
+ .17 <5 (i- 0(1):: 0‘“ 1).: >1 (121)
U 0 O
_ 1 ' - 1
‘ 6nm [2 £2 szkz 2 x12 + Y An] ' (122’

where, for convenience we defined

2
Y z biz (V-v ) _ 2b12‘12 + 1.x
‘ 52‘ 12 U 2 12 ° (123)

Details of the above calculation are found in Appendix A7.
Before proceeding with the calculations it is worth

rewriting (118), taking in account (119) and (122):

54

8 Big (A +A ) ~ 2
E U n 9. ~ 3 H ~ b12
n,£ e [<"lfix— In> ('0‘ + Y) (An'Az) +
~ hi2 -1 ~ afiH ~
+ kT g (Ed) " T An) <nI5—f— I$><$IHIn>] = 0 . (124)

Notice that the constant terms of (122) (specially the

long range Coulomb interaction terms Z vkkkfi) have been

' 2

b
cancelled out. Since now we know that —l3;+ Y) is second

U
3H
order in A , we just have to evaluate <fiI-a-A—Hlfi> to the

lowest order in A). Straightforwardly we find

~ 2
a b
. a - _ 1 - ~ (1) (1) ~ ~ _ 12
<nl—U In> — 2 Z <noIT Q Ni+Ni+ Q Tlno> — ——2 An (125)
U 1 U
and
35 2b
H ~ 2 ~ + (1) ~ ~ 12
<n|—— In> = - - Z X<n (c- c. Q Tln > = - —A ~(126)
abl2 U <ij> o o 10 30 0 U n

The second set of terms inside the square bracket in
(124) has an overall factor kT assumed to be higher order
than A . Then we only have to calculate these terms up to

order of A2 . With (110) we can rewrite it as

2 b
b12
¢ U

'1 I?“ I" ~I l
A ) <5 ¢><¢ H fi>
n a (127)

-1

3H
“2 ~ .11. " (1)
A ) Cn <no|(l U T Q )

5T§|$><$|n(1-%0(1"i)|fio>.

55

A careful examination of (127) shows that within the

second order in A , the only eigenfunctions |$> that

enter in the calculation are the ones resulting from the
removal of the degeneracy of the first excited state of

Ho , with the same number of electrons NS , which lies higher

by U . If we then express |$> in terms of |$o> , as

we did in (110) for In> and Ifio> , we have

 

 

I$> = 5¢ [1 - (P + 0(2)) :—l:— (P + 0(2))T]Q(l)l$o>
H -e
o 1
= 0¢ (1 - s i 0(1))|$ > (128)
where
s s 9 ~ 1~ p + Q(2) ~ 1~ 0(2) . (129)
Ho-el Ho-sl

The normalization constant C and energy E are

¢
found to be

b2 -1
52 = [1 + —lZ-(B + B')] (130)
0 U2 ¢ c)
b2
E = u + -13 (B - 23') (131)
¢ U ¢ ¢
where
B¢ E 22 Z <$Olc10 cjO P 030' Cio'|$o> (132)

<i,j> 00'

56
B$ E 2) X <~ lo? (2) +
<i,j> 00'

Then we proceed as in Appendix A8, and get the final form

of the stationarity equations (124)

b2
8 02(AH+A2)bi2 biz
HQ, 9 {T An(An-A)L) (T + Y) +
b b ~
12 .12 _ _ _ —_221§1
+kTU_2_An [U (V V12) tizl} “0" an
and
b2
12
B U (A“+A£) b12 biz
nize {:fi"An(An'Az’ (-6— + Y)
(133)

 

b ~
'1 _12 _ _ _ _ 3F[H]
+ kT U Ant U (V V12) t12];} ‘ ° ‘ ablz °

From these two equations it follows easily that

2
b
12 _
—E— + y — 0
b
.12 _ - _ . .

(the Am, defined in (117) and (113), are not needed!).
Recalling the definition of Y in (123) we find from

(133') that the best parameters b12 and U are:

(1 + ~13) (134)

57

 

U = (V-v ) (1 + —l3) (135)
12 K
12
where
2
K = _ 2t12
12 V-v12

is the exact nearest neighbor kinetic exchange; Kij was

defined in Eq. (45). U = w is seen to be a solution of
(133'), which on physical grounds can be seen to lead to
higher free energy.

It can be seen from (110) that In> is a linear com-
bination of singly—occupied-site functions Ifio> and
functions <ij> g C10 cjolfio> which have one doubly-occupied
and one empty site and all the other sites are singly-
occupied; the ratio of this mixture is easily seen to be
b12/U . If a similar perturbation theory is done on the
exact Hamiltonian H , the same sort of mixture in the
wavefunctions is found, the ratio of the mixture being equal
to tlz/(V - v12) . Our results (134) and (135) for the
best parameters show that the minimal principle (43) matched

the wavefunctions of both the exact and trial Hamiltonians,

since

b12/U = t12/(V - V12) . (136)

58

Another important feature of our results (134) and
(135) is that the low-lying eigenvalues of both Hamiltonians
have been matched; in other words the appropriate Heisenberg

exchange integrals are the same (despite the lack of Ui' in

H):
210i2 2ti2
— = — = K O
U (V-vlz) + v1221 12 + X12 (137)

Hence we have accomplished Anderson's objective of defining
a non-magnetic one-electron Hamiltonian whose transfer
integral leads via the kinetic exchange to the correct
low-lying exchange splittings when IK12I>>X12.

However, the above approach fails to give a physically
acceptable approximation to the properties of H when

0

12 > |K12| .5 This is seen as follows. We assumed, in

the course of making our calculation, that U>>|b12l>>kT ,

X

leading to (134) and (135); but when X12>IK (135) gives

12| .

U<0 , which is inconsistent with our initial assumption.

The only alternatives to that assumption are 0<U ~ Iblzl ,

0<U<<Ib12| , or U<0 . It can be seen that any of these

assumptions would lead to a very poor approximation to H .
In section V.C we discuss an improved model for the

special simple case of two sites.

59

3.3 Two Sites; All Temperatures

 

The results so far derived for narrow-band systems
show that the best Hamiltonian of the form (79) is tempera-
ture dependent, the low temperature one being appreciably
different from the high temperature one. Unfortunately,
we are not able to find how to connect the best low tem-
perature parameters with the best high temperature ones,
because the expansion techniques we have used do not hold
for this intermediate region of temperatures. However,
since our results (in the grand canonical ensemble) are
independent of the number of sites, we decided to study
numerically the simple case of two sites only, for all
temperatures, with the hope of extrapolating the inter-
mediate temperature results for an arbitrary number of
sites. This also could provide a check on the analytical
expressions for the best parameters. I

The exact single-band Hamiltonian (3) specialized

for this two-site case is explicitly:

H = h11(N1 + N2) + V(N1+N1+ + N2+N2+’ + v12N1N2

' _ _ + +
+ g [h12 + v12 (N10 + N20)](c10c20 + C20 C10)

1 +_ _ +_ _ + +
+ X12 g [2'(clo 020 + c20 Clonclo c20 + C20 Clo)

-N N

10 20] (138)

60

where some new symbols are introduced:

v12 5 V1121 (139’

X12 3 v1221 '

The eigenstates of (138) are easily found and they are dis-
played in Table 1. The eigenstates of the Hubbard
Hamiltonian for two sites can immediately be drawn from that
same figure just by replac1ng h11 by b11 , V by U , h12 by

. _ . _ _ . .
b and setting v12 — v12 - X12 — 0 . The eigenfunctions

12
are the same except for two singlets.

The variational procedure consists of calculating
the trial free energy F[H] (left side of (43)), carrying
the traces over the eigenstates of the Hubbard Hamiltonian
(exactly) and minimizing F[H] as a function of bll’ b12’ U

' I
for given sets of hll' V, V12, v12, X 2, h12' u and T .

1
Figure 1 shows the output of one of the calculations as
function of the temperature. The input parameters were
chosen to simulate a small overlap situation. The numerical
calculation not only checked the analytical results but

also showed how to connect between low and high-T results.
Presumably, for a larger number of sites, the behavior is
similar. We noticed that for smaller overlap the flatness

of the low-T region extends to higher kT/IK (still kT<<It

12l 12')“

For very low temperatures the free energy is insensitive

to a change in bll , which agrees with our previous discussion

61

(sec. V.B.2). However, if we make the numerical calculation
coming from higher temperatures to lower ones we notice that

the best b tends toward a limiting value which is that

11
required to give the same number of electrons for both
Hamiltonians.

We concluded before that the minimal principle (43),
in establishing the best trial Hamiltonian HH tries to
match up the eigenfunctions and the low-lying splitting
in both H and HH(b).

the low-T regime. Here these results have been confirmed

These conclusions were established for

numerically. In fact, examining Table 1, it is easy to see
that, for small overlap, C (the coefficient which mixes

doubly-occupied with singly-occupied states in the singlets)

~

is the same for both H and HH(b), with the results (134)

and (135), namely blz/U = tlz/(V-v the eigenfunctions

12);
are then the same. It is even easier to see that the
low-lying splittings are equal, in the small overlap regime:

just compare (A1.ll) with (A1.12), using (134) and (135).

Figure l.

62

12, U and b11 for the best

half-filled Hubbard Hamiltonian are shown as

The values of b

functionsof kT, in units of V. Also is
shown the ratio b12/U° The parameters in
the exact Hamiltonian were, in units of

V: V = 0.2, h = 0.05, V' = 0.03,

12

= X12 = O .

12
(so t

12

= 0.08), h

12 11

b../v
0100

0.090

0.080

LCD
0.90

030

b./V

0.30

0.20
0.0

bu!“
' 0.l0

0.09
0.08

63

 

 

 

 

 

I

 

 

I

 

64

C. Narrow Half-Filled Band; The "Best" Modified Hubbard

 

 

Hamiltonian (With Potential Exchange)

 

As we showed in section V.B.2 our initial approach
with the trial Hamiltonian being the Hubbard Hamiltonian
is successful in one sense and a failure in other. It
is successful in accomplishing Anderson's objective of
defining a non-magnetic Hamiltonian which leads through
the kinetic exchange to the correct exact low-lying
exchange splittings. But it fails in the sense that it
cannot give the correct physics of the exact Hamiltonian
when the potential exchange exceeds the magnitude of the
exact kinetic exchange, this suggesting that adding an
effective potential exchange to our trial Hamiltonian
might resolve this difficulty. We therefore propose
studying the full (56) as our trial single band Hamiltonian.
We have begun an investigation of this for the simple case

of two sites.

C.1 Two Sites; Low-Temperature Limit

 

 

The Hamiltonian (56), which we now call H specialized

HX

to the two-site case is

65

:11:

+ N N ) +

= b11(N1 + N2) + U(N1+Nl+ 2+ 2+

HX

+
+ C

20 +

C

+
b12 g (C10 C20 10)

+ +_ __
U12 g (Clo C20 C20 C10 NIONZO) (14°)

(still rather simpler than the corresponding "exact"
Hamiltonian (138)). Its eigenstates are easily obtained
from Table l by replac1ng V, hll' h12' X12 by U, bll’ b12

and U and setting v12 = viz = 0 , except for the singlets

12

which are shown separately in Table 2. This exception occurs

C +

C+
20

because of the lack of terms (cI- c 10.

o 25
- - + c ) in H
C20 C10 C20 10 HX '
Again, we have to solve the stationarity equations (54),

these for one more variational parameter taken as U = U =

12 21

U12 . So far we have studied this problem in the low
temperature region only, so it will be discussed only in this
regime of temperature. If we make the restriction on the
traces that we did for the previous low temperature calcula-
tion (for an arbitrary number of sites), namely taking the
traces only over the eigenfunctions originating from G , we
will not be able to determine variationally as before, the
parameter bll; further, we will not be able to find uniquely
the other parameters because the inclusion of the effective
potential parameter U12 destroys the independence of the

stationarity equations. We then felt it was necessary to

66

extend the space in which the traces are to be calculated,
to include the states for which N = NS i 1 (originated
from E ). In other words, for this particular case of
two sites, the traces are taken over the lowest singlet,
the three triplets (N = NS = 2) the four states for which
N = NS - l = l and the four states for which N = N3 + l = 3.
The fiHX- uN eigenvalues corresponding to these eight
states lie above the ground level by roughly U/2 .
In evaluating the traces in the stationarity equations
) in

we have expanded F'i (singlet eigenvalues of fiHX

the following way:

4132
fi" = % [(U + 012) - ((U — 012)2 + 16 hi2)312012 - —513
4b2 (141)
"'+ =-§- [(0 + 012) + ((U - 012)2 + 16 hi2)"]=u + —%13 .

The quantity U12 , as we pointed out before is expected
to be second order in the overlap.

After a long but straightforward calculation we end
up with the following form for the equations BFIHHx]/8A

where A = bll' U12, U and b12’ respectively:

(2v ' 2b + 2h + U - X

12 11 11 12 - U+V)cosh Bb12-2v

12 Slnh Bb12 = 0

12

(142)

67

and

-h + 3L+

-263 -866 —203 _ _
3e L + 2 e i} [(2(U V) + b11 v12 11

_ _ - - ' "
+ U X12) cosh Bbl2 (3b12 3t12 v12) Sinh BblZ]

12

_ .. .. _ I - =
+[(bll v12 h12+x12 U12) cosh Bb12+(b12 t12+v12) Sinh Bblzi} 0

(143)

and
~ 3b2 ~ ~
e"28J -—%3 L + e-28J(3 + e'ZBJ) 5% blz M +
U U
2 2
- ~ 4b 8b
+e 280 e ZBJ [( ——l3 (U-V+U -x ) - L(1———12) +
2 12 12 2
U U
+ (v -b +h -U+V) + kT 8b M) cosh Bb +
12 11 ll 32 12 12
8bi2
— - __ _ ' °
+ ((b12 t12)(1 U2 ) v12) Sinh Bblz] +
— — - ... ' . =
+3[(v12 bll U+V) cosh 8b12 + (b12 t12 v12) Sinh Bb12]} 0

(144)

68

and

2
-28J 6b12
e ——-—

U

L . e-ZBJ (3 . ...-sz ....ng

) M +

8b

 

-gsU -283 _ _ 8kT 12 _ _
16b12 b .
+ T L) cosh Bblz + (16 1—[21-(b12-t12)+(U-V+U12-x12))SlnthlZJ

I
O

I

+3['2(b12-t12)COSth12 + (U-‘«'+U12-X12)sinh Bblz]?

where we have defined

 

[I E U 4‘1312
4b b
: 12 _ _l§ _ _ —
L ' "6"[b12 2t12+ U (V V12)] 2“’12 X12)
b
- 12
M : ffi—'(V-V12) ' tlZ
~ 216?2
J = U + U12

In the above equations we have replaced b11 - u by
(-U + U12)/2 which fixes the number of electrons in the
system equal to two.

In this regime kT<<|b12I; then we replace sinh 8b12

by s cosh Bb12 , where s is the sign of b The

12 '
percentage error made with this replacement is of the order

(145)

(146)

(147)

(148)

(149)

69

of exp -28]b12I which gets drastically smaller when kT
gets lower and lower. With this replacement we obtain from
(142) an expression for U12 - X12 in terms of the other

quantities appearing there. We found it convenient to

insert that expression into (143) - (145) wherever U - X

12 12
appears explicitly. Then keeping, inside the curly bracket,
terms which are zero and first order in the overlap we see

that the system of equations (142) - (145) reduces to a.

much simpler one

U-V + bll-hll-v12 ~ S(b12-t12-v 12) = 0 (150)
Egla [b - 2t + b12 (V-V )] - U + X - 0 (151)
U 12 12 —fi_ 12 12 12 —
b
12 _
T (V‘Vlz) - 1:12 (152)
2V12 - 2b11 + 2h11 - U + V - 28v12 + U12 - X12 = 0 .(153)

It is convenient to note that Eq. (152) implies that b12

has the same sign as t12 (since U>0) .

Solving then the above system of equations we get

 

2v v
12 12
U=v+—__—|t l—-—_—— K (154)
V
_ v _ 12
biz ' V-v t12 S V-v K12 (155)

12 12

70

 

_ _ 12
U12 ’ x12 V-v K12 (156)
12
V
b = h + v - s v' - —£3— It | (157)
11 11 12 12 v-v12 12

to second order. These results hold for any ratio Xlz/IK12

 

Because of the complexity of the equations (142) - (145),
we have not been able to estimate analytically the order
of magnitude of the error made in determining the best
parameters. Presumably it is smaller than second order in the
overlap. We made a numerical calculation which agrees with
the values found above. Although for each best parameter the
leading order would be satisfactory, we have to know them up
to second order in overlap, in order to solve for 012 (which
is essentially second order in the overlap). See for instance,
equation (142) or (153). We note the consistency of our
original assumptions,.U, bll’ b12' U12 are respectively
zeroeth, zeroeth,~first and second order.

With the results (154) - (157), it can be seen that the
wavefunctions of the exact and trial Hamiltonians have again
been matched (compare C and C' in Tables 1 and 2,
respectively, in the limit of small overlap). Also the
low-lying splittings have been made identical to second order

~

in overlap; in other words, the Heisenberg exchange JHx

~

appropriate to H is equal to J

HX appropriate to H

H

~

_ _ 2 _ _
JHx — 2b12/U + 012 — K12 + X12 — JH (158)

71

To check it see (Al.13), (Al.11) and (154) - (157).

The chemical potential necessary to fix the number of
electrons in the best fiHX equal to two is, according to
(Al.10), equal to

hfl<

. (159)

In Appendix A1 we calculate the chemical potential to

make the exact Hamiltonian H have two electrons; and for
kT << (Itlzl - (Vizll I ItlZI > Ivizl (160)

the expression (159) above is the correct one to second order.

For the other situation, namely

kT << lltlzl - Ivi2I| I Itlzl < (Viz) (161)

the exact chemical potential is not the same as in (159),
differing from that in terms which are first order in the
overlap. The former case where ltlzl > [viz] is the more
realistic one at least for the hydrogen molecule.

Also for the conditions (160), the separation in
E - uN between the ground state of the system (a singlet)

and the lowest of the (NS i 1) levels has been matched

perfectly (to second order): This separation GHX for
~ (b) . U _ _3_ _ 4132 h
HHx is 3 2 U12 |b12| + 52 as seen from t e

72

right hand column of Table l; the corresponding separation

+

 

. . . _ v _ _3_ _
for H , under the condition (160) is 6 — 5 2 x12 |t12|
4t§2
. From (154) - (157) it can be seen that to second
V-V12 '
order,
6 = 6 - (162)

It is interesting to notice that the lowest of the

(NS i 1) levels is made up of bonding states, for t12 < 0 ,

or antibonding states, for t12 > 0 ; since the sign of

b12 is the same of t12 , correspondly the same happens

with the Hamiltonian Hg?) . Now with (161) the degeneracy

of this level is between bonding states for (NS 1 l) and
antibonding for (NS 1 1), depending on the sign of viz and
, - ~(b)
t12 , whereas in the HHX
above (there being no degeneracy between bonding and anti-

the situation is the same as

bonding states). This would appear to be a reason why the
results are better for (160) than for (161).

But in any case the leading term in the expression for
the best U (namely U = V) matches the position of the
(NS 1 1) levels with respect to the ground state, to zero
order in overlap. Further, the position of the highest
levels, namely those with NS and (Ns 1 2)partic1es (all of
which are degenerate for HHX-uN) matches the average

position of the respective levels in the exact model. See

Figure 2 to clarify these points.

Figure 2.

73

~

The eigenvalues of i) H - uN , ii) HH - uN

and iii) HHX - uN in the zero and small

overlap limits for the half-filled-band
grand canonical ensemble. In case i) it is

assumed that t 0 , v' < 0 and

12 < 12

It and kT << ||t - Iv'

12| > lviz' 12| 12II °

The levels in ii) and iii) were drawn

roughly as in the best HH and fiHX .

74

.fl'l
6"?)
2
4t
2vl2’+2Iv'2l+2X'2+ —-'-2 4M2;
_2 2 V"'I2 ‘u"
+ —-.°=
-1- )('2'+I12’l+4lv I
"2" ”I2 2 I2
55",

v if? 'I2+2xI2+"I2' u lbl
"2'"I2°;- "2'+ I2
'2 'I2"'2XI2" l'I2""2"'I'2'

-11."
2 ~13.
"2"”(2'
-V- =v.2+2Iv"2I '22 -U 2
—v- -:-v— v 2g--+2III2I4-2x|2v-‘_—-"'|2 203.1312

.9. .L
2 "' 2 uI2"’."’I2|

I
""2""'2"’I2"“’I2l

75

C.2 Effective One-Electron Potential

It is interesting to try to find an effective
one-electron potential W(r) such that

<w.|K + wlw.> = b.. (163)
1 J 13

where K E p2/2m is the kinetic energy operator. The most
interesting case is in connection with our best Hamiltonian

~

HHx containing the effective potential exchange. Since we
have so far found this Hamiltonian only for two sites our
discussion here will be only on the effective potential

for this case.

The diagonal element of (163) namely b11=h11 + v1212 ,
to the leading order in the overlap can be reproduced if one
uses for W the Wigner-Seitz potential st . This is
defined as the potential "seen" by an electron in a
Wigner-Seitz cell. For two sites, the Wigner-Seitz cell
associated with an ion is the half-space from the mid-point
of the site axis, including that ion. In this case, st
is the potential due to the singly positive ion in the cell

that the electron finds itself, plus the potential due to

the other atom. We can write then that

3

wi(£f)d r'

 

<w. IW IW > = Z I _ ( I-
1 WS k j C811 j E B‘j gfj —I_r_ £1

1
.TE:E;T)] wk(£) . (164)

76

With (164), the diagonal element of (163) is

 

 

 

2 2
_ 3 _ e e
<w1|K + Wfis|w1> — f d r w1(£)(K )wl(£)
all space I£_Bll '12"le
3 2 "3(E')d3r'
+ f d r w1(£) f (165)
cell 1 all space Igfg'l
w (3')
+ f d3r wi(£) f d3r' 1
cell 2 all space Irfir'l
which to zero-order in the overlap becomes
2 2
2 2 r w (r)w (r')
<w1|K - —EL———-— -—£L——-|wl> + f d3r I d3r' 1 ’ 2 ‘ =
‘ LIE-51' '12"le |£-£' '
= h11 + v1212 = b11 ' (166)

If the Wigner-Seitz potential is used to calculate
the off-diagonal element of (163) it does not reproduce

our best b12 given by (155):

<w1|K + wwslw2> = K + 6

12 (167)

12

where

K12 s <w1|K|w2> (168)

77

 

and
912 : <w1|""w|"'2> '
2 2
= - d3r w1(£)w2(_r_) e - f d3r wl(£)w2(_r_)—e———
cell 1 _r_-§1| cell 2 Ig-gzl
2
2 W n
--l d3r w1‘£""2‘£’ [_3_ _ ez d3r. 2(r)
cell 1 gfgzl all space Iger'l
3 e2 2 wi‘E'MBr'
- f d r w1(r)w2(r)[————-— - e ] (169)
cell 2 — _ {-51 all space IE'E'
which to the lowest order in overlap is
3 e2 3 e2
612 = - d r wl(£)w2(£)—————— — I d r w1(£)w2(£)——————.(170)
cell 1 lgfgll cell 2 £~§2
Our best transfer integral b12’ to lowest order in the
overlap, is
V V I e2 I
b = —:-—- t = —_—-— {K - <W ——-— w >}
12 v v12 12 v v12 12 1 Ir-Rll 2
= K12 + Y12 (171)
where
2
Y12 = V—_-v {V12K12 " V<W1| |W2>} (172)
12 rfigll

<wl|

78

So the difference between (167) and (171) is 612-712

which is equal to

 

 

 

v 2
12 e 3
w -W|w>=_ [2f w(_1;)w(_r_)-———dr-K]
WS 2 V V12 cell 1 l 2 IE—Bll 12
V I e2 3
+ _ [ W (9W (E) ———-—- d r
V v12 cell 2 1 2 |£7§1|
e2d3r
- I W1(£)w2(£) ]
cell 2 lg-gzl

which does not appear to be zero in general. In fact it seems
to be of first order in the overlap, and therefore it is not
expected to be negligible, although we have not made a
numerical estimate.

That an improvement over the Wigner-Seitz potential
is needed can be seen intuitively when one realizes that
for small overlap most of the hopping integral comes from
the region where the electron position r is in between
the ions. For the WS potential, the "other" electron
discontinuously jumps from one ion to the other as the
electron of interest (at r) crosses the midpoint (which
seems unreasonable). The quantity t12 is easily seen
from eq. (31) to be the matrix element between w1 and w2
of K and the potential due to the nuclei plus one electron

2

distributed equally over the two ions (in w1 and wg), a

much more reasonable picture. However, the factor

79

V/(V - v12) increases the difficulty of finding explicitly
the corresponding potential W (which we have not been able

to do), this W appearing to be a complicated one and a

function of the kinetic energy.

VI. SUMMARY AND DISCUSSION

A new approach to the problem of narrow-band magnetic
semiconductors has been presented and investigated in the
context of the half-filled single band problem. The
approach consists of using the minimal principle of
statistical mechanics for the free energy, to find a
"best” Hamiltonian H, containing one-electron terms as
well as certain important two-electron terms, but much
simpler than the exact Hamiltonian. It is important to
realize that if H consisted of only a one-electron term,
the thermal Hartree-Fock approximation would have been
obtained from the variational principle, as the best
one-electron Hamiltonian, which has been proved to have
its failures for the narrow-band problem.

It has been recognized that one-electron approxima-
tions are definitely bad and that two-electron Coulomb
interactions are very important for narrow-band electrons.
Hubbard and Anderson, working not inconsistently with this
view, presented theories in which one-electron-approximation

terms (Hartree-Fock) are to be used in combination with

80

81

important two-electron terms, to make physical predictions.
Although on the right track, their theories were shown to
be unsatisfactory (sections II and III), essentially
because they made a one-electron approximation first and
then combined the resulting one-electron term with
two-electron terms. Instead, we find the variationally
best Hamiltonian H with both types of terms: the best
one-electron term is found simultaneously with the best
two-electron terms. The investigations discussed in this
paper in the context of the single-band model showed that
our approach overcomes the difficulties cited in the
previous theories.

In particular we studied two cases. In one, the trial
Hamiltonian was taken to be of the form of the Hubbard
Hamiltonian, and in the other, effective potential exchange
terms were added to it. In the first case the low-T
results matched the low-lying exchange splittings appropriate
to both trial and exact Hamiltonians despite of the lack
of potential exchange terms in H . Hence we accomplished
Anderson's objective (not attained by him) of defining a
non-magnetic one-electron Hamiltonian, whose transfer
integral leads via the kinetic exchange to the exact
low-lying splittings to leading order in the bandwidth.
However this first approach fails to give a physically
acceptable approximation to the properties of the exact

Hamiltonian when the potential exchange dominates the

82

kinetic exchange. In the second case, explored so far in
the simple case of two sites, the same matching of
low-lying splittings is obtained, but for any_ratio of
potential exchange to kinetic exchange, which certainly
is an important improvement over the first trial
Hamiltonian. Further, the centers of gravity of the

(NS 1 1) - electron levels are matched to zero order in
overlapSI. (The first trial Hamiltonian erred in zero
order for those levels).

As far as the near future of the exploration of
this new approach is concerned we have at least three
immediate problems to consider. The first is the
generalization of the results of the best Hubbard
Hamiltonian augmented by the potential exchange, to
an arbitrary number of sites. The second is connected
with the problem of superexchange, where one has
closed-shell ions and non-closed—shell ions (MnO, for
example). Here we hope to find that the appropriate
set of Wannier functions (the best in our procedure)
will achieve the rapid convergence of the perturbation
theory originally desired by Anderson. The third has
to do with adding higher cation bands; our approach
applied to a model containing such bands should enable
one to take into account the quantitatively important

35

"electron rearrangement" effect , as it modifies the

one as well as the two-electron parameters.

83

Finally, we would like to point out that a rather
remarkable suggestion concerning the broad-band or
high-density limit results from our narrow-band con-

10, the restricted HF approxi—

siderations. As is known
mation for the electron gas leads to the incorrect
result that the density of states is zero at the Fermi
energy, and the source of this error is the long range
of the Coulomb interaction via the exchange integral.
Thus it would seem that the addition in H, of a short
range interaction (like the Hubbard intra atomic term)
to the one-electron operator would not be able to
overcome this fundamental difficulty. However, this
View may be quite incorrect. We essentially saw

(Sec. II) that the restricted HF approximation led to a
hopping integral bi'

J
narrow-band region, the source of the error again being

that behaved very badly in the

the long range of the Coulomb interaction via the
exchange contribution to the HF eigenvalues. But, as we
saw, the addition in H of the Hubbard intra atomic
interaction was quite sufficient to overcome the diffi-
culty found in the narrow band region. Thus the
possibility that it might similarly overcome the diffi-
culty in the region of large ratio of bandwidth to

intra atomic interaction no longer appears to be

an unlikely one. It is under investigation.

APPENDICES

APPENDIX A1

TWO-SITE SINGLE-BAND MODELS

The exact single-band Hamiltonian Specialized for

the two sites case is

+ N N ) + v N N +

H = h11(N1+N2) + V(N1+N1+ 2+ 2+ 12 1 2

. _ _ + +
+ g [h12 + v12 (N10+N20)](cloC20 + cZoclo) +

+ + + +
+ x12 g [3(C16026 + C25C15)(Cloc20 + CZOClo) ' N1oNzo] (A1°1)

where

v12 5 v1212

' :
v12 _ v1121 (Al.2)
x12 5 v1221

This Hamiltonian can be straightforwardly diagonalized, the
eigenfunctions and the eigenvalues being shown in Table 1.
There we also display the H - uN eigenvalues for the special
case of <N> = two electrons (in average) in the system.

The chemical potential which fixes the number of electrons

equal to two is determined by the equation:

tr e-B(H-UN)(N-2) = 0 (Al.3)

84

85

or

] e -

%B(V+2v -X ) B(t -v' ) -B(t -v' ) Ba
1 + e 12 12 SE? 12 12 + e 12 12

B(t +v' ) -B(t +v' ) 38a 48a
[e 12 12 + e 12 12 1 e ;} - e = o (Al.4)
where we have put
X
_ y, _ 12
u — h11 + 2 + v12 —§— + a (Al.5)

and a is now to be determined. The solutions for some

special interesting cases can be immediately found:

1) kT >>It12|+lvi2| a = o (Al.6)
It12|>|V12| “ = “sviz

2) kT<<||t12| - Ivizll It |<IV' l _ -8't (Al.7)
12 12 a ‘ 12

where s and s' are the signs of t and V12 , respectively;

12
in other words t12 = Sltlzl and viz = s'lvizl .

The other model considered in this paper is the Hubbard
Hamiltonian whose eigenstates can easily be drawn from the
Table 1 just by replacing V,h11,h12 respectively by
X = O . The chemical

U,b and setting v

... I =
11'b12 12‘V12 12
potential that fixes the number of electrons equal to two

is easily found to be

_ E
u — bll + 2 (Al.8)

86

for all temperatures.
Another model Hamiltonian of interest, used in section

V.C is

+ N N +

1+N1+ 2+ 2+)

HHX = bll(N1+N2)+ U(N

) + (Al.9)

+ + + +
b12 g (clOCZO+C20Clo U12 g (CloCZOCZBClo NloNZC)
Its eigenstates can also be obtained from Table 1 by making
the appropriate replacements, except for the singlets, which
are listed separately in Table 2.

The chemical potential to fix the number of electrons

equal to two is

U12 (Al.10)

_ E _
u ‘ b11 + 2 2

The low-lying singlet-triplet splitting is listed below
for the three two-site Hamiltonian models considered in the
text; in each case we first show the splitting calculated
exactly and next the approximation for small overlap:

For H, this splitting is (E )

triplet - Esinglet

2
4t
_ -1 - _ 2 2 % : __l£_.-
v12 2x12 2 [(V+v12) ((v v12) + 16 tlz) 1 - V-v 2x12 (Al.11)

12

87

~

while for HH

-%—[U—(Uz+16b2)"];——

12

~

finally for HHx , the splitting is

1 2
- U12 - §[(U+U12) ((U-U12)

+ 16 b

2
12)

8

]

2

4b
:__12_
-0 20

12

(Al.12)

(Al.13)

Table l.

88

Listed are the eigenstates of the exact
Hamiltonian H, (138), and E - uN for
half-filled band condition; a is a function
of temperature and other parameters in H

(See Appendix Al.). Here,

2 2

Bi = (1/2) [(V+v ) : ((V-v ) + 16t

12 12

c = (v - E-)/2t A = (1 + C2)-% ,

12 12'

1
= I = _ _ _.
t12 h12 + V ' 5 2 V v12

n = 3h11 + V + 2v12 - X12 , la > = (c c

+ + + +

+ +
°2+°1+H°> ' Ia2> “ (C1+C1) + °2+°2+)|°> '

89

 

 

 

 

 

 

 

|> H|> = EI> E - uN
|0,0> = [0) 0 0
- = 2
I1 ,+> 2 (c1+-c :+)|o> h t + g
V' t1 +V'
_ _% c+ 11 12 1212 12
> =
4.
ll ,+> = 2 l5(cl++c 2+)|O>
+ - ' -
1+ +> _ 2_% h11 t12 V12 C+t 12 v12 C
' ' ‘ (C1+C 2+)l0>
[2-0> = A[ a > - C a > - - - -
. l 1 l 2 1 2h11+X12+E v 2v12+2X12+E -2a
lzfo> = A[Ia1> + C|a2 >1 2h + +
11 X12+E+ -V-2v12+2X +E -2a
12
I2 O> = 2 h( + c+ c+ 0+ > - -
_ C+ +
|2'++> ‘ C1+ C2+IO>
_ c+ +
'2'*+> ‘ C1+ C2+'°: 2h11+V12x 12 'V’V12'2C
2 ++> = 2;5 c* +
I ' (C1+C 2+ C2+ C1+)|°>
3- +> = 2";5
I ' (C1+ C2+)C 1+ C2+“)> ”_t v. .
I3 +> = 2_%(C+ _ + ) + + 12 12 C‘tlz'V12'3C
' 1+ C2+ C1+C2+lo>
+
'3 '1) = H%(C ++C2+)C1+C 2+|°>
n+t +v' + +
I3+ +> = ”C(c c+ ) + + 12 12 5 C112 V12 30
' 1+ C2+ C1+C2+'°>
_ + + + +
|4,o> — c1+c1+c2+c2+|o> 4hll+2V+4v12 -4a
-2x12

 

 

 

Table l

90

~ +
. Here E" =

Table 2. The singlet eigenstates of HHX

2

(1/2)[(U + U + ((U — U12)

2 8

~

C' = ( - fi'—)/2b and 3' = (l + C'2)-15 .

U12 12

91

 

 

 

 

 

Singlet H I) = E. >
Eigenfunctions HX
+ + + + + ~,—
A [(C1+C 2+ + C2+C1+) C (C1+C1+ + C2+C 2+)]|°> 2b11 + E
+ + + + + + + + ~.+
A [(C1+C 2+ + C2+C1+)+C (C1+C1+ + C2+C 2+)”0> 2b11 + E
-2 -
(C1+C 1+ C2+ C2+"0> 2b11 + U

 

 

Table 2

 

APPENDIX A2
NATURAL CONDITIONS FOR A UNIQUE CHOICE OF WANNIER FUNCTIONS

IN THE CASE OF SINGLE BANDS.

The Bloch functions, in terms of the atomic functions

a(£f§n) are as follows:

-;5 i£.3n
TE(£) = (NSFE) g e a(£é§n) (A2.l)
where
13.31“
Pk E 2 e Aim)
— m
A(n) a 1 51(5) a(£-§n) d3r = A(-n) . (A2.2)

We have assumed a(£) =a*(£)=a(I£]) .
Let S be a unitary symmetry operation of the crystal
such that 551 = 3i. ; then one can show with (56-a) and

(A2.l) that

-'k-R.+iy
-% 1— —1 k
Sw (1;) = N‘ 2 e — \v (1‘)
Bi 5 k SE
= N 2 e —-W (r) (A2.3)
s k E_—

In particular let S be inversion through an origin at the

lattice p01nt: in which case SEi = Bi = O and Ys‘lk = Y-k .

92

93

And further, demand that w .(r) be invariant under S

(as the atomic function):

This requires
n (A2.4)

where nk is an integer.

If we demand the Wannier functions to be also real, as

are the atomic functions, we must have

= 2m n (A2.5)

where mk is an integer.

From (A2.4) and (A2.5) it follows that

= R n (A2.6)

with 2k being another integer. However, demanding that the

Wannier functions go to the atomic functions as the overlap,
(A2.2) for n # 0 , goes to zero we finally get:

Yk = constant-

APPENDIX A3

SOLUTIONS Cfi‘ EQUATIONS (60) and (61)

Because of (63) and (64), the thermal averages in equations

(60) and (61) are very simple to evaluate. For instance,

2 <N.N. N. > can be written as
j 1 3+ 3+

Z tr pO NiNj+Nj+ = 2 tr pO NHNi+ + tr p0 NiNj+Nj+ —
= 2g + (tr 5.N.) 2 tr 6.N. N.
1 1 . . + +
3#1 J J J
= 2g + fi(N-1) 6
where
g = tr 50 Ni+ Ni+ = tr 51 Ni+ Ni+
and
fi = tr 5 N = tr ~ N .
o 1 pl 1

These quantities g and a can be easily calculated with the

help of Table 3. It then follows that

“B(Zb -2u+U),
a = e 11 /E

and

-B(b 'U) -B(2b -ZU+U)
fi = 2(e 11 + e 11 )/§

94

(A3.1)

(A3.2)

(A3.3)

(A3.4)

(A3.5)

95

where

“B(b ‘U’ "B(Zb -2U+U)
E = l + 2 + e 11 + e 11 .

In the same way we did for (A3.l),we can show that
eq. (60) and (61) can finally be written in terms of g

and n as

.1[2§ + fi(1-fi)] = o

(U-V)(n-2)g + [hll-b + n X V1j13

1 .
1 J#1

and

(U-v)(l-§) + [hll-bll + fijgl vljlj](fi—2) = o .

From these two equations we find

(U-V)f(fi,~) = o
and
[h -b + fi 2 v . .]f(fi,§) = o
1 .
ll 1 j#l 1313

where

f(fi,§) = fi(1-fi) + 5(35—2) ~2§2 .

If f(n,g) = 0 , U and bll will be undetermined
although there will be’a relation between them. Let us
study (A3.11). Assume f(fi,§) is zero and solve for S in

terms of 5 . The solutions are n = l+§ and fi = 2g .

From (A3.4) and A3.5) n can never be equal to l + g ;

(A3.6)

(A3.7)

(A3.8)

(A3.9)

(A3.10)

(A3.11)

96

and also n can only be equal to 2g if n = a = 0 , which
of course is of no interest here. Hence flfi,g) # 0 and

(65) and (66) follow immediately.

97

Table 3. Eigenstates of fil and fil - uNl . The last

mud

column is for u = bll + (half-filled band).

98

 

(al-u~1)|> = (El-uN1)|>

 

 

 

 

; El El uNl E1 -(b11+%)N1
|o> o o
|+> bll bll - u - u/z
Ii) b11 b11 ’ “ ‘ ”/2

|++> 2b11 + U 2b11 - zu + U 0

 

 

 

 

Table 3

 

100

Straightforwardly we can show that

~ + + +
[H0. ckO(S)] = U Nk0 ck0(s) - u Cko(s) (A4.8)
[RO, c£0(s)] = -U NkO c£0(s) + u c£0(s) . (A4.9)

Then we can rewrite (A4.5) in the following form

3

SE Ak£o(s) = BU<Nk3 - N23) Ak£0(s) (A4.lO)
which after being formally integrated from zero to
5 gives
SBU(Nk5 ‘ N25) +
Ak£o(s) = e , Cko C20 (A4.ll)

which proves (A4.l).

APPENDIX A5

SOLUTION OF EQUATION (101)

Taking into account (77), (78) and (79), and neglecting
terms which are higher than first order in the overlap,

we can write (101) in the following way:

1
+
g Io dx[(hji-bji)<gij5(x) Cio Cjo Cjo Cio>o +

+ +

+ ijij<9ija<x> Cio Cjo (Ni5+Nj3) cjo Cic>o
+ +

+ k;i,j vjkik (gijo(x) Cio CjO Nk Cjo Cio>o]

l x
+
+Bg bji Io dx {yo ds [<gij5(s) cjo Cid Cio Cjo d>o -

+ +
—<gij5(X)gjio(s) Cio cjo Cjo Cio d>o]
‘Il d5 < ’( ) '(S) + c+ c d>

o gijo X gjio Cjo Cid i0 jo o

+<g .-( ) c+ c c+ c ><d> = 0 (A5 1)
le x jo id id jo 0 °
where
_ l '

d = X (hkk - bkk) Nk + 3 2 Vk£k2 NkN2 (A5.2>

k k,£
We have used the fact that gjiE(X) gij5(X) = 1 by the
definition (93), and also that U = V.

101

102

To evaluate the thermal averages in (A5.l) we recall

the fact that 5 = H 5 , where
o k k
~ -B[fi - uN ] ~
0k = e k k /z (A5.3)
~ _ ~ - -B[Hk - uNk] (A5.4)
z z z - tr e
k
and Hk = bkk Nk + U Nk+Nk+ ; (A5.5)

and because (99), (100), and (63), for half—filled band we

have

fi - UN

k = V/2 [-N + 2N

k kf Nkil . (A5.6)

k

The traces in (A5.l) should be taken over the eigenstates

of fi - uNk , which are shown in Table 3. But before doing

k

that, it is convenient to write:

= Vijij[NiNj - (Ni+Nj)] +kgi jvkikiHNiijNk - 1)- 2 Nk]
+ 2' v [(1/2) N N - N ] . (A5.7)
. . 1 2 2
kl£#1lj k k k k
We have considered 2 v . . = 2 v . . . Above, 2
k#i.j klkl k¢i.j kaJ k.2¢i.j

means sum with k#i,j and £¢i,j and the prime means k#£ .

103

Then eq. (A5.l) becomes

1
X f dx(hji — b.. + Z

+
v. . )<g..-(x) c. c. c. c. > +
o o 31 k#i,j jk1k

130 10 30 30 10 o

+ +
ijij< gijo‘x x) Cio Cjo (N13 + NjS) Cjo Cio>ol +

l
_ + +
+ 8 gb ji Iodx'{}ijij If: ds (< gijo (s) ch ci0 ci0 cjO

(N.N. — N. - N.)>
1 j 1 j o

+
-<gijo(x) gjio(s) Cio Cjo Cjo Ci0 (NiNj - Ni - Nj)>o)

l
+ +
_I odS<gij3(x)gji5(S)Cjociocioc jo‘Ni Nj - Ni - Nj)>o

+
-<gji5(X)CjoCiocioc jg>o]

+

_[2 z .Vkiki + 1/2£ {#1 j vkzkfil If: ds(< gij5(s)c jg Ciocioc jO>o _

+ +

‘(X) g --(s) c. c. c. c. > )

- <gijo jio io jo jo i0 0

-f:d < (x) -( ) c+ c+c >
S gijoX gjig S jocio 1oC jg o

+ +

+ <gjia(x) CjOCiOCiOC cjo>013 = 0 (A5.8)

The next step is then to evaluate the thermal averages

and then do the integrals. As an example we do the first one:

104

1
X 1 dx <g..-(x) cf c. c. c. >
o o 130 10 30 30 10 o

xBVN.- -xBVN.-

+
= Z I dx <e 10 N. > <e 30 c. c. >
o o 10 o 30 30 o

l 1 L—
(2/22) I dx (e’z8V + eXBV) (1 + e<2 X’BV)
0

8V
(4/22) (eléBV + EEVZl) . (A5.9)

Similarly we calculate the others and finally find the result

given in (102).

APPENDIX A6

PROOF THAT fieff IN (114) CORRESPONDS TO A HEISENBERG

HAMILTONIAN

The operator fie in (114) can be written as

ff

” = - _ii + + =
Heff Z. X . U F[Ciocjocjo'cio'1P
13 00
2 (A6.l)
— - 2' 2 Eli P[N — N N - c+ -c+-c ]P
— ij 0 U i0 i0 jg iocio jo jg

The P operator can be dropped if we remember that fieff is
to operate on the ground level only; in other words on
states containing only one electron on each site.

The relationsbetween the above creation and destruction

. . 29
operators W1th the sp1n operators are6’ :
+ — S — S +'S
Ci+ci+ ‘ 1+ ’ ix 1 iy
+ .
c. c. = S. = S. -18.
1+ 1+ 1- 1x 1y
1
§*Ni+ Ni+) ‘ Siz '
S1nce in our case Ni+ + Ni+ = 1 then
_ l
Ni+ — f + Siz
_ l _
Ni+ ‘ i Siz

105

106

With (A6.2) and (A6.3) in (A6.1) we can write

2
b..
.. _ _ 1 _ _ _ _
Heff - i5 —U-J— [1 (15+Siz)(35+5jz) (L. s12) (15 sjz)
- si+sj_ - si_sj+] =
= ' Z. U [:5 _ 2§-i°—S-j] '
l]

The eigenvalues En' (115), of EH calculated through
second order degenerate perturbation theory, differ from
the eigenvalues of the Heisenberg Hamiltonian with

__ _ 2
Jij 2bij/U, by the constant

I
2
Nsbll + Ej bij/ZU -

APPENDIX A7

CALCULATION OF EQUATION (121)
Taking the expression for the exact Hamiltonian given
in (3) we have for the first term in (121)

<molHIno> — 5 Efi Vk£k2<molNkN£|no> +
(A7.l)

[no >- -<m OlNkonlfiofl ,

Vk22k[ mo IC ”kg 20 C26Ck6
The other terms in H give no contribution to this matrix
element; their sequence of creation and destruction operator
are unable to, starting from |fi0> with only singly-occupied
sites, give back another function |m0> with also only
singly-occupied sites. The first term in (A7.l) is clearly

I ~ = ~ ,
Z vkfikkanm because NkN£|n0> Ino> , the second term,
k2

NIH

C+ [5
koc 20 C20 Cko
the curly bracket, can be written as

after adding and subtracting <mo |c+ o> inside

c+

l '
7 Z vk££k[§0 (m o'CkoC 20 C20 Cko'

|fi0> - 6 ] . (A7.2)
k2

nm
The potential exchange integral Vklik is a quantity of the
order of the square of the overlap between k and 2 . It's
clear that overlap between non-nearest neighour sites is

smaller than nearest neighbour overlap; then neglecting

107

108

terms which are higher than second order in the nearest

neighbour overlap we can then write (A7.2) as

 

X X X <m [C c c .c Jn > — — X 6 . (A7.3)
12<k£> 00. 0 kg 20 £0 kg 0 2 12 nm
Be a s 2' + c c+ is ro t'o 1 t
H = Pi 0(1)&P of which [E > is an ei nf nctio we
eff U o ge u n
then get
~ ~ _ l ' _ 1 l ] .
(molHlno> ‘ 6nm ['2' 1E2 Vk2k2 5: X12 + 5' X12 An (“'4’

The second matrix element in (121) can be written as

2b [ X X h.. <m

. C. . . In >
2 . . ] I I
l < > . I O 10 |O IO 10 O

X Z v <mO|NiE+N n > + (A7.5)

+ .. ii'i ' ' '
<13) 00' j j 1o 30 30 10

~ + + ~
+ . . < N . _ . ' . > .
<§j> £#i,j Vlkjk g0. mol k clocjoc30 CIO'Ino ]

. + + ~ . . .
It 18 clear that ciocjocjo'cio'lno> 15 an e1genfunct1on of
N13, Nja and Nk , with eigenvalues 0,1 and 1, respectively

(k#1,j). Then it follows that

A 6 . (A7.6)

(1)~ ~ =
H O Tlno> b12t12 n nm

109

The third term in (121), after neglecting higher

order terms, becomes

(1) (l)~ ~ _ 2 ~ + + ~
Tlno> _ b12V X 2 X <molciocjoNkak+cjo'Cio' no>

<2 1
mOIT Q . . '
<1,j> k 00

H Q

+ <m Ic+ + ~

2 I
b Z V X X . c. N N c. .c. ,In >
12 k£ k£k£<ij> oo' o 10 30 k R 30 1o 0

_ 2 2 1 '

12 n Cm <ij> oo' 2k,£#ij
x <m |c+ + ~

. c. c. .c. ,In >
o 10 30 30 10 o

] A 6 . (A7.7)

l
0"

Finally, we obtain the result shown in (122).

APPENDIX A8

CALCULATION OF TERMS IN EQUATION (127)

With (127) and (128) we can write

a
<nl— H|Y><Y|Hln>

BHH
3A

(1)

<nMI(l-— 0(1)) _(1"STQ(1))[YO>><YO|(1‘— Q T)Ino > .(A8.1)

And for A=U this becomes

{-l <fi

l (1)
U 0

Q(l)TInO > + l <§O|TPH|fiO>]} +

~ ~ ~ ~ 1 ~
TQ Ni+Ni+lYo>[<YolH|no> -U (YOIH U

+ higher order terms. (A8.2)

Evaluation of (A8.2) is similar to the ones we did before,

as we see if we rewrite it in the form

_1, ~ (1) (1) ~ (1) (l ) (1) ~ ~
UmaTo Nmuo “%>+UTJTQ NHN.N Ho Thy
_1 ~ (1) (1) ~ ~ ~
U2 <noIT Q Ni+Ni+Q T P Hlno> -
Then it becomes
2
b t b
12 12 12
[ U + —55 (V v12)] An , (A8.4)

111

and the whole second set of terms in (124) is, for A = U:

 

b b
12 12

For A = b (A8.1)turns out to be

2 <fi [of c I? >[<? |H|fi > — l <? IT P H n > -

<.. o 10 jo o o 0 U o o
13>
(A8.6)
- l <~ In o‘l)'1'~|fi >1 = -[b———12(V-v ) - t 1 A
U Yo 0 U 12 12 n '

and the whole second set of terms in (64) is, for A — b ,

b

kT 12
l—fi-(V v

-U_ 12) — t12] , (A8.7)

LIST OF REFERENCES

10

11

12
13

14

REFERENCES

J. C. Slater, Phys. Rev. 81, 385 (1951); 83, 538 (1951).
P. Hohenberg and W. Kohn, Phys. Rev. 136 B, 864 (1964);
W. Kohn and L. J. Sham, Phys. Rev. 140 A, 1133 (1965).
L. Hedin and S. Lundqvist, Solid State Phys. 23, (1969).
F. Seitz, Modern Theory of Solids (McGraw-Hill Book Co.,
New York, 1940) p. 329-

J. Hubbard, Proc. Roy. Soc. A 216, 238 (1963).

P. W. Anderson, Solid State Phys. 14 (1963).

By this term we mean the standard thermal Hartree-Fock
approximation. For the special case of zero-temperature, see
e.g., ref. 8; and for the general case see ref. 9.

J. C. Slater, Quantum Theory of Molecules and Solids

(McGraw-Hill Book Co., New York, 1963) Vol 1.

For a recent review see A. Huber in Mathematical Methods

 

in Solid State and Superfluid Theory, Scottish
Universities Summer School (1967) (Ed's R. C. Clark and

G. H. Derrick), Plenum Press, N.Y. (1968).

J. Bardeen, Phys. Rev. 50, 1098 (1936).

C. Herring, Magnetism, Vol IV (ed. by G. T. Rado and

H. Suhl, Academic Press, New York and London (1966))

p. 310-311.

T. A. Kaplan, Bull. Am. Phys. Soc. 13, 386 (1968).

T. A. Kaplan and P. N. Argyres, Phys. Rev. B1, 2457 (1970).

R. Bari and T. A. Kaplan, Phys. Rev. B6, 4623 (1972).

112

113

15 - The conclusion that the HFA is unsatisfactory for such

16

17

18

19

20

systems has been reached by Slater, Mann, Wilson, Wood,
Phys. Rev. 184, 692 (1969), sec. IX, on grounds which
apparently differ from those of ref. 12 but really are
very similar. Namely, it was noted by Slater (reference
8 secs 5-2, 6-2) that atomic excited states which con—
serve the number of electrons are not given at all
correctly by the HFA. Whereas Kaplan's point (ref. 12)
that the zero-point entropy is incorrect in the HFA in
the atomic limit, is essentially the statement that the
low-lying spin-flip excitations in a small magnetic field
(which conserve electron-number) are given very badly by
the HFA(even for a hydrogen lattice in the atomic limit).
See J. C. Slater, T. M. Wilson, J. H. Wood, Phys. Rev.
119, 28 (1969).

J. C. Slater, Phys. Rev. 165, 658 (1968).

Despite wide study of this method in atoms (see, e.g.
ref. 16 and 17).

They are actually overcomplete as can be seen from the
simple exactly soluble two-site two-electron Hubbard
model.

This contrasts with the broad-band or high-density
situation where the interactions presumably can be
treated perturbatively in a calculation of the
single-particle propagator. In this case the fs(x) which
grow out of the non-interacting set f:(x) will have large

norms (~1) whereas the others will have small norms.

21
22

23

24

25

26

27

28

114

It is these "large" fs(x) that are apparently taken

(ref. 3) to correspond to the results of a band calcu-
lation.

L. Matheiss, Phys. Rev. 23! 3918 (1970).

See J. C. Slater et al, of ref. 15.

For more recent work along these lines see D. J. Newman,
J. phys. c _6_, 2203 (1973).

Nilton P. Silva and T. A. Kaplan, Bull. Am. Phys. Soc. 18,
450 (1973); Nilton P. Silva and T. A. Kaplan, AIP Conf. Proc.,
No. 18, Magnetism and Magnetic Materials (1973), p. 656.
J. Kanamori, Prog. Theo. Phys. 10, 275 (1963);

M. Gutzwiller, Phys. Rev. Letters, 19, 159 (1963);

Y. Nagaoka, Phys. Rev. 111, 392 (1966);

D. I. Khomskiy, Phys. Metal Metallor. 12, 31 (1970);

E. H. Lieb, and F. Y. Wu, Phys. Rev. Letters, 20, 1445 (1968);
A. A.Ovchinnikov, Sov. Phys. JETP 10, 1160 (1970);

D. Cabib, and T. A. Kaplan, Phys. Rev. B1, 2199 (1973);

R. A. Bari, and T. A. Kaplan, Phys. Rev. B6, 4623 (1972);
and many others; to find other refs. see H. Shiba, and

P. A. Pincus, Phys. Rev. B5, 1966 (1972).

This model has been used quite commonly, e.g. by Slater

(ref. 27) for H and by Mattheiss, Phys. Rev. 123, 1219

2
(1961).
Reference 8, Chapters 3 and 4.

E. C. Kemble (1937), "The Principles of Quantum Mechanics",
Sec 480, McGraw—Hill, New York. (Reprinted by Dover,

New York, 1958).

29

30

31

32

33
34

115

T. A. Kaplan, unpublished lecture notes for a course in
Magnetism given at Michigan State University in 1971
and 1972.

The exponential behavior of wi(£) was found under the
assumption of zero phases Yk of the Bloch functions

(see App. A2).

Also see W. Kohn, Phys. Rev. 115, 809 (1959) for a similar
result in the case of one-dimensional bands of arbitrary
width.

Since Hubbard's explicit intent was to consider a model
appropriate to transition metals, where presumably it

is essential to have a broad band (45) cross the narrow
band (3d), our proof that Hubbard's derivation leads to
grossly incorrect results is perhaps unfair because we
consider an isolated s-band (more appropriate to an
insulator). However Hubbard did apply his model to
insulators (Proc. Roy. Soc. A 181, 401 (1964)). In any
case, it is important to realize that this type of
Hartree-Fock approach fails.

See discussion in Kaplan and Argyres, page 2457 of ref. 13.
In particular, Kaplan (ref. 12) noted that in the thermal
HFA the entropy + 0 as T+ 0 for zero bandwidth as compared
to the exact result Nk 2n (28+1), where S is the ionic
spin; this is essentially the reason for the fact that as
T+ 0 the spin-susceptibility x+ 0 in HF whereas the exact

result is x+ ot=(the Curie law). These show that the

35
36

37

38

39

40

41

42

43
44

45

116

non-zero-T HFA gives results that are vastly different from
the picture in the Heisenberg model, where in the atomic
limit Jij+ 0 , so that there are N non-interacting spins.
See T. A. Kaplan, AIP Conf. Proc. No. 5, Magnetism and
Magnetic Materials (1971) p. 1305.

N. Fuchikami, J. Phys. Soc. Japan 88, 871 (1970).

It is interesting to note that t12 is the average of the
two biz (+,+).

Fuchikami (ref. 35) gave unique values for biz (+,+);
she presumably considered biz (+,+).

Furthermore, this 0 dependence has been shown explicitly
(T. A. Kaplan, unpublished).

Nai Li Huang Liu and R. Orbach, AIP Conf. Proc. No. 10,
Magnetism and Magnetic Materials (1972) p. 1238.

K. I. Gondaira and Y. Tanabe, J. Phys. Soc. Japan 81,
1527 (1966).

Hubbard, Rimmer and Hopgood, Proc. Phys. Soc. 88, 13
(1966).

Petros N. Argyres, T. A. Kaplan, Nilton P. Silva,

Phys. Rev. 88, 1716 (1974).

Hund's rule was implicitly assumed in ref. 21.

S. V. Tyablikov, Methods in the Quantum Theory of
Magnetism, (Plenum Press, New York, 1967) p. 196.

~ -2aRij 'aIE-Bil
Actually vijji ~ in a Rij e for ai(r) ~ e
'“Rij v

in which case Aij ~ e ; thus vijji = 0(Aij )where v

is less than but arbitrarily close to 2 and it is in this

sense that the term second order is used. See ref. 13.

46

47
48

49

50

51

52

117

R. S. Tu and T. A. Kaplan, Physica Status Solidi (b) 88,
659 (1974).

R. A. Bari, Phys. Rev. 88, 2662 (1971).

See also D. J. Klein, W. A. Seitz, Phys. Rev. 88, 2236
(1973).

It can be seen that the second order corrections to the
wave-functions omitted in Eq. (109) contribute to

Eq. (118) only in higher order than the terms we keep.
That this might occur in situations of physical interest
has been noted in ref. 39.

Under certain conditions (|vi2I<lt12I), the location of
the bottom of the (NSi1)-electron levels relative to the
ground state energy is matched perfectly to second order
in overlap.

It is interesting to note that the free energy variational

(minimal) principle leads to information concerning

excited states, (and therefore dynamics), even as T+ 0.

174 8092

03