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I. The Electronic and Geometric Structures of
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Jeffrey Lee Tilson

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I . THE ELECTRONIC AND GEOMETRIC STRUCTURES OF
VARIOUS PRODUCTS OF THE SC+ + H20 AND H28
REACTIONS.

II. THE ELECTRONIC AND GEOMETRIC STRUCTURES OF
+ScSe AND +ScSeH

By

Jeffrey Lee Tilson

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1993

ABSTRACT

I . THE ELECTRONIC AND GEOMETRIC STRUCIURES OF
VARIOUS PRODUCTS OF THE Sc+ + H20 AND H28

REACTIONS.

I I . THE ELECTRONIC AND GEOMETRIC STRUCTURES OF
+ScSe AND +ScSeH

By

Jeffrey Lee Tilson

The products of the Sc+ + H20 and H28 reactions were
investigated by constructing ab-initio MCSCF and MCSCF+1+2
wavefunctions for various states of +ScL, +ScLH, Sc+LH2, HSc+LH and

H2"-+ScL (L=O and S). The energies were computed at the optimized

geometries. A Mulliken population analysis was performed for each
molecule. Where possible, comparisons to experimental data are
made. The ground state product of the Sc++H20 reaction is the
insertion product, H-+ScOH which assumes a cis conformation and is

40 kcal/mol below the reactants. The two reaction products,
H2"'+SCO and H-+ScOH, are nearly degenerate (AB = 5 kcal/mol) and

are both the result of an exoergic reaction. The H2"°+SCO product is
the ground state +ScO molecule electrostatically bound to H2 and is
35 kcal/mol below the reactants. The ground state of the reaction
Sc+ 4- H28 is the H2--°+ScS electrostatic species (34.5 kcal/mol below
Sc+ + H28 ) while the electrostatic product, Sc+SH2, is exoergic by only

11.4 kcal/mol. The insertion product, H-+ScSH , was examined and is

not a minimum on the Sc+ + H28 reaction surface.

The 12+, 3A and 32+ states of +ScSe and the 2A and 223+ states
of +ScSeH were analyzed. The ground state +ScSe is a triply bonded
species of 12““ symmetry with a bond strength of 84 kcal/mol. The
3A and 32+ excited states lie higher in energy at 31 and 28 kcal/mol,
respectively. The +ScSeH molecule has a 2A ground state nearly
degenerate with the excited 22"’ state with both differentially
stabilized by formation of the Se-H bond. This stabilization is

consistent with the work on +ScOH and +ScSH.

To Red, Joan, John, Nana and Herb

iv

ACKNOWLEDGMENTS

I first want to sincerely thank Dr. James F. Harrison for
accepting the job of being my mentor. His intellectual support,
guidance, patients and stamina throughout my time in graduate
school was second to none. No less important are the contributions of
my guidance committee. In particular, Drs. George Leroi, Dan Nocera,
Ned Jackson and John Allison for their advice and opinions. Thanks
also to the faculty and staff of the Department of Chemistry (MSU)
for help, advice and encouragement when it was needed and
especially to Dr. K. Hunt and Dr. T. V. (TVA) Atkinson, for allowing
me to frequently interrupt them to discuss the occasional intellectual

curiosities of mine.

My time in graduate school would have been much less
significant without the contributions of the expert service facilities

and, of course, my friends.

Finally, I wish thank Michigan State University and the
Department of Chemistry (MSU) for financial support during my

graduate career.

TABLE OF CONTENTS

Page
LIST OF TABLES ............................................................................................. ix
LIST OF FIGURES ........................................................................................... xii
KEYS TO SYMBOLS AND ABBREVIATIONS .......................................... x v
CHAPTER I
INTRODUCTION ................................................................................... 2
CHAPTER II
THE ELECTRONIC AND GEOMETRIC STRUCIURES OF
VARIOUS PRODUCTS OF THE Sc+ + H20 AND H28
REACTIONS.
INTRODUCTION ................................................................................ 5
BASIS SETS ....................................................................................... 6
FRAGMENT ENERGIES ................................................................... 7
MOLECULAR CODES ....................................................................... l O
AB-INITIO CALCULATION OF +ScO AND +ScS .................... l 3
A. GENERAL CONSIDERATIONS ....................................... 1 3
B. MCSCF RESULTS FOR +ScO ............................................. 1 5
C. MCSCF RESULTS FOR +ScS ............................................. l 8
D. MCSCF+1+2 RESULTS FOR +ScO 2 0
E. MCSCF+1+2 RESULTS FOR +ScS ................................... 2 l
ELECTRONIC DISTRIBUTION .................................................... 2 3
A. +ScO ...................................................................................... 2 3
B. +ScS ....................................................................................... 2 6
C. COMPARISON OF +ScO AND +ScS ............................... 2 8

vi

Page

D. DIFFERENCE DENSITY CONTOURS ............................. 3 O
AB-INITIO CALCULATIONS OF +ScOH AND +ScSH .......... 3 2
A. +ScOH .................................................................................. 3 2
B. +ScSH ................................................................................... 3 7
C. COMPARISON OF +ScNH, +ScOH, AND +ScSH ........ 4 1
REACTION PRODUCTS ................................................................. 4 2
A. H—+ScOH ............................................................................ 4 2
B.+&£&h ............................................................................... 45
C. COMPARISON WITH THE +Sc +N H3 SYSTEM ...... 4 8
D. +Sc +SH2 ............................................................................ 5 1
E. H2-+ScS .............................................................................. 5 4
F. H-+ScSH .............................................................................. 5 5
CHEMISTRY-PREDICTION S ....................................................... 5 7
CONCLUSIONS ................................................................................ 5 9
REFERENCES ................................................................................... 6 6
CHAPTER III

THE ELECTRONIC AND GEOMETRIC STRUCTURES OF
+ScSe AND +ScSeH

INTRODUCTION .............................................................................. 7 1
BASIS SETS ...................................................................................... 7 2
MOLECULAR CODES ...................................................................... 7 2
FRAGMENT ENERGIES .................................................................. 7 3
MCSCF FUNCTION FOR +ScSe ..................................................... 7 4
MCSCF+1+2 FUNCTION FOR +ScSe ............................................ 7 9
ELECIRONIC STRUCTURE OF +ScSe .......................................... 8 2
+ScSeH GENERAL CONSIDERATIONS ....................................... 9 1

vii

Page

EUKHRONKXHRUCHHHHXHSdkH ....................................... 95
THfimmKXfiflfiflSflnf ..................................................................... 101
flflmMARY ......................................................................................... 106
IGTEMDKES ...................................................................................... 108
APPENDIX A
EUKHRONKHHRUCHflGHflEDRfiTEGmflQUES
DURODUCHON ................................................................................ 111
HAMHJONUDI ................................................................................ 112
CONSTRUCTION OF THE WAVEFUNCTION .............................. 1 l 6
SCFEQUATKDJ .................................................................................. 124
IDUENQONS ....................................................................................... 126
REHHUDKIS ......................................................................................... 131
APPENDIX B
LIST OF PUBLICATIONS ................................................................ 1 3 4

viii

LIST OF TABLES

Table Page
HAPTER II

1. Total Fragment Energies (au) ....................................................... 11

2. Dissociation Energies (kcal/mol) .................................................. 1 2

3. +ScS and +ScO Equilibrium energies (Emin’ au), Vibrational
frequencies (we, cm'l), Bond Lengths (re, an) and

Dissociation Energies (De, kcal/mol) ...................................... 1 9

4. Valence J’ScO(H) MCSCF orbital

populations at equilibrium ........................................................ 2 4

5. Valence +ScS(H) MCSCF orbital

populations at equilibrium ......................................................... 2 7
6. MCSCF Gross atomic charges ........................................................... 2 9
7. +ScS(H) and +ScO(H) Equilibrium energies (au), Equilibrium
bond lengths (au), and Dissociation energies

(kcal/mol) ............................................................................................ 3 9

8. Summary of MCSCF+1+2 Sc+ + SH and OH fragment
energies (kcal/mol) .......................................................................... 4 4

ix

Table Page

9. H2+SCO and H-+ScOH Equilibrium Energies (au), Bond
Lengths (au) and Angles (deg) ................................................... 4 6

10. 3A2 and 3A1 states of Sc+SH2. Optimized Geometries and

Total Energies ..................................................................................... 5 3

11. H2---+ScS Optimized geometry, Total Energy and

Dissociation Energies ........................................................................ 5 6

l2. MCSCF+1+2 Dissociation Energies (De, kcal/mol) ..................... 6 2
HA R 11

1. Fragment Total Energies (au) ......................................................... 7 5

2. Dissociation Energies (kcal/mol) .................................................... 7 6

3. +ScSe, +ScS, and +ScO Equilibrium energies (Emin’ au),
Vibrational frequencies ((06, cm'l), Bond Lengths (re, au)

and Dissociation Energies (De, kcal/mol) ............................... 8 0

4. Valence +ScSe(H) MCSCF orbital

populations at equilibrium .......................................................... 83

5. Valence +ScS(H) MCSCF orbital

populations at equilibrium ......................................................... 8 6

OO

\0

Table

Valence +ScO(H) MCSCF orbital

populations at equilibrium .........................................................

MCSCF Gross atomic charge distribution .................................

+ScSe(H), +ScS(H), and +ScO(H) Equilibrium energies (au),
Equilibrium bond lengths (au), and Dissociation

energies (kcal/mol) ........................................................................

Summary of MCSCF+1+2 Sc+ + HZSe, HZS and H20

fragment energies (kcal/mol) ....................................................

xi

Page

87

89

93

94

LIST OF FIGURES

Figure Page
HAPTERII

1. MCSCF potential energies of the 12"”, 3A, and 32+
states of +ScS and +ScO relative to the

ground state asymptote .................................................................. l 6

2. MCSCF+1+2 potential energies of the 12.", 3A, and 32+
states of +ScS and +ScO relative to the

ground state asymptote .................................................................. 2 2

3. MCSCF Total density contours (TDCs) and Difference
density contours (DDCs) for the 123+, 3A and 32+ states

of +ScO ..................................................................................................... 3 l

4. MCSCF Total density contours (TDCs) and Difference
density contours (DDCs) for the 223+ state of +ScOH and
the12+ and 32"" states of +ScO ....................................................... 3 4

5. MCSCF+1+2 relative energies (kcal/mol) ..................................... 3 6

6. MCSCF Total density contours (TDCs) and Difference
density contours (DDCs) for the 22+ state of +ScSH and
theIZZ+ and 32+ states of +ScS ........................................................ 4 3

xii

7.

8.

9.

1.

Figure Page

Illustration of the energetics amongst the Sc+ + 8H2

reaction products. Minima on the curve are MCSCF+1+2

optimized values ................................................................................ 5 8
MCSCF+1+2 relative energies of the Sc+ + 0H2 reaction ....... 6O
MCSCF+1+2 relative energies of the Sc+ + 8H2 reaction ........ 61

HAP R III

MCSCF potential energies of the 122+, 3A and 32+

states of +ScSe, +ScS and +ScO relative to the

ground state asymptote .................................................................. 7 8
MCSCF+1+2 potential energies of the 12+, 3A and 32+

states of +ScSe, +ScS and +ScO relative to the

ground state asymptote ................................................................... 8 1
Radial distribution function of the ground state +Sc (3D)

Inner ls core orbital not displayed ............................................. 8 5
Orbital populations of +ScL and +ScLH (L=O, S, Se) .................. 9 0

xiii

Figure Page

5. MCSCF Total density contours (TDCs) and Difference
density contours (DDCs) for the 22‘."‘ state of +ScSeH and
thelil+ and 32‘." states of +ScSe ..................................................... 9 7

6. MCSCF Total density contours (TDCs) and Difference
density contours (DDCs) for the 22‘." state of +ScSH and

the12+ and 32+ states of +ScS ...................................................... 9 8

7. MCSCF Total density contours (TDCs) and Difference
density contours (DDCs) for the 222"” state of +ScOH and
the12+ and 32* states of +ScO ...................................................... 9 9

8. Contours of the valence natural orbitals for the 22+ states of

+ScLH at equilibrium ........................................................................ 1 0 0

9. MCSCF+1 +2 relative energies (kcal/mol) of selected
Sc+ + SeH2 products ........................................................................ 1 0 3

10. MCSCF+1+2 relative energies (kcal/mol) of selected
Sc+ + 8H2 products ......................................................................... l 0 4

11. MCSCF+1+2 relative energies (kcal/mol) of selected
Sc++ 0H2 products ......................................................................... 105

xiv

KEY TO SYMBOLS AND ABBREVIATIONS

Symbol or
Abbreviation Mug
SCF Self-Consistent-Field
GVB Generalized Valence Bond
MCSCF Multiconfiguration SCF
CI Configuration Interaction
MCSCF+1+2 Single and double excitations
from a MCSCF reference space
SCF+l+2 Single and double excitations
from a SCF reference space
‘I’ Wavefunction
¢ Spatial or spin orbital
% Electronic hamiltonian
fl Antisymmetrizing operator
V7- Laplace operator
97 F Fock Operator] Matrix
]j(l) / 7(5-(1) Coulomb/Exchange operator
Jij / Kij Coulomb/Exchange energy
< ij l ij > /< ij lji > Coulomb/Exchange energy
au Atomic Units
Energy = hartree
Distance = Bohrs (0.527 Ang.)
<||> Bra-Ket Notation
Re or Rmin Equilibrium distance
Be or Emin Equilibrium energy

XV

Symbol or
A r vi i 11

Xu
PX, Py, pl

dxz'l'y2'222. dxy. dxz, dyz. dxz‘y2

TIE
III:

xvi

Meaning

Vibrational frequency (cm'l)
Electron index
Nuclear index

Configuration State Function

Spin functions (spinors)
Atomic number of the nth nucleus
Orbital (one-electron) energy
Kronecker Delta
uth Basis function
Atomic (Real) p functions
Atomic (Real) (1 functions
Total density contour

Difference density contour

CHAPTER I

CHAPTER I

INTRD N

The focus of the work presented in this dissertation is
electronic structure calculations of transition metal (TM) containing
species. The techniques employed are Multiconfiguration self-
consistant-field (MCSCF) and configuration interaction calculations
(CI). These methods are significantly more advanced than SCF
calculations and are required to incorporate the important near
degeneracy effects present in all TMs.

The work presented in Chapter II examines the possible
products of the Sc+ + H20 and H28 reactions. Experimental data
available for the Sc+ + H2O reaction are compared. No experimental
data are available for the SH2 reaction. Many structural similarities
exist between these two reactions, but the ground state reaction
products are different.

Chapter III continues with a theoretical examination of
+ScSe(H), +ScS(H), and +ScO(H). The ligands (O, S, and Se) belong to
Group VI and therefore have the same valence structure. This
analysis allows us to make predictions for the reaction of Sc+ + SeH2.
The filled 3d-shell of Se is found, as expected, to not significantly
influence the bonding structure of +ScSe(H) relative to +ScO(H) and

+SCS(H).

3

Appendix A is included to outline the theoretical techniques
used throughout this dissertation, with particular attention paid to
the Hartree-Fock (HF) wavefunction and its extensions.

A listing of publications resulting from this work is presented

in Appendix B.

CHAPTER II

CHAPTER II

THE ELECTRONIC AND GEOMETRIC STRUCTURES OF

VARIOUS PRODUCTS OF THE
Sc+ + H20 AND H28 REACTIONS.

INTRD ON

We are interested in characterizing the possible products of the
gas phase reaction of the monopositive ions of the early transition
metals with H201 and H2S.2 In this dissertation the focus is on the
simplest of these ions, Sc+, and this chapter reports the results of ab-
initio electronic structure calculations on the systems +ScL in their
12+, 3A and 32+ states, the +ScLH 2A and 22‘." states and the reaction
products H-"’ScLH, H2+ScL, and +ScLH2 where L=O and S.

There has been extensive experimental and theoretical work on
the reactions of transition metal ions with small ligands. Pertinent to
this work are the +MO (M=Sc, Ti, etc.) bond strengths3 (in particular
+ScO Do: 159:7 kcal/mol) and the several experimental +M-OH and
+MO-H (M=Sc, Ti, V, Cr, etc.) bond strengths.4 Recent results indicate a

+Sc-OH bond strength of 87.8 kcal/mol and further suggests the
reaction of Sc+ with H20 yields the product Sc+---OH2 with an

interaction energy (D0) of 31.4 kcal/mol.5 There are no experimental

data available for the +ScS(H) species.

6

The results of this work indicate that the bonding of H to the O
in +ScO and the N in +ScN6 causes a differential strengthening of the
Sc+ to oxygen and nitrogen bonds of approximately 43 kcal/mol.
This strengthening results from a ligand to metal sigma dative bond,

formed in concert with the bond between N or O and the H atom.

This added stabilization causes the 80"" + H20 ground state reaction
product to be the insertion species,1 HSc+-OH, (AB = -40 kcal/mol)
with the electrostatic species, +Sc-"OH2, slightly higher7 (AB = -36.2

kcal/mol). The analogous J'Sc + SH2 reaction products were analyzed

with emphasis placed on the strength and structure of the induced

sigma dative bond.
The point groups employed in all calculations are either C2v or

C This is possible because all the studied species are of the same or

S.
higher symmetry. It is always possible to use a less discriminating,
i.e. Lower, point group since it will fully contain all operations of the
higher point group. This lowering of the point group order in a
calculation will increase the amount of computational time but yields

exactly the same results. The similarity of results allows us to

interpret orbitals in terms of atomic type orbitals regardless of the

selected point group. Throughout this dissertation the atomic, C2v

and Cs orbital symmetries will be used interchangeably.

BAI ET

The Scandium basis set used in this study consists of the

(l4s,9p,5d) basis from Wachters8 augmented with two diffuse p

7

functions (Dunning)9 and a diffuse d function as recommended by
Hay.10 This set was contracted to (58,4p,3d) following Raffenetti.11

The Oxygen basis was the (lls,7p) set from Duijneveldt12
augmented with a single diffuse d (exp=0.85) function and contracted
to (4s,3p,ld) following Raffenetti.“

The Sulfur basis was the (12$,9p) set from Huzinaga13
augmented with a diffuse 8 (exp = 0.60), a diffuse p (exp = 0.04) and
a diffuse d (exp=0.3l) function and contracted to (55,5p,1d) following
Raffenetti.1 1

Two basis sets were used for the Hydrogen atom. The first
consists of the Huzinagal3 (4s) augmented with a single p (exp=l.00)
function and contracted to (2s,1p). This was the set chosen for the
+ScOH, H-+ScOH, +ScSH and H-+ScSH calculations. The second basis set
consists of the above 45 basis augmented with a single s (exp=0.03)
function and three p (exp=1.00, 0.33, 0.11) functions. This set was
contracted to (3s,3p) and used in the H2°--+ScO and H2--'+ScS

calculations. This basis was previously shown to adequately

represent the polarizability of the H2 molecule.14

FRAGMENT ENERQES

Sc+

The ground state15 (3D, 3d14s1) energy was computed using
the SCF and SCF+l+2 (substitutions from only valence electrons)
functions. The Sc++(2D, 3d1) SCF energy was also determined. The
total energies plus the energy for the mixed state Sc+(3B2, 3d813d1t1)

are collected in Table l.

The Sulfur 3P state15 was analyzed with MCSCF and

MCSCF+1+2 wavefunctions. The MCSCF function was constructed
from the in-out correlation (GVB) of the doubly occupied 3p1rx orbital
plus all valence spin couplings. This results in 5 CSFs of 3B2

symmetry. All valence single and double substitutions (of 3B2

symmetry) from the MCSCF reference space result in the 619 CSF
MCSCF+1+2 function. These energies are collected in Table 1.
SH

The 2H state of SH was examined by a 2B1 17 configuration

MCSCF function. This was constructed from all spin couplings of a
GVB(2/4) function (correlating the 1: bond and the doubly occupied S

3px orbital) and a 3,766 CSF MCSCF+1+2 constructed from all

y
valence single and double substitutions (of 2B1 symmetry) from the

MCSCF reference space. The total energies are listed in Table 1 while
the dissociation energy (De) is collected in Table 2.
H28

The energy and Optimized geometry of H28 was computed with
a 37 CSF MCSCF function constructed to correlate in a GVB way the
two bonding orbitals and the S doubly occupied out-of—plane orbital
plus all spin couplings and a 23,922 CSF MCSCF+1+2 function derived
from all valence single and double substitutions from the MCSCF
reference space. The equilibrium energies are listed in Table 1 and
the dissociation energies in Table 2.
O

The Oxygen 3P state15 was analyzed with MCSCF and

MCSCF+1+2 wavefunctions. The MCSCF function was constructed

9
from the left-right correlation (GVB) of the doubly occupied 2p1cx
orbital plus all valence spin couplings. This results in 5 CSFs of 382

symmetry. All valence single and double substitutions (of 3B2

symmetry) from the MCSCF reference space result in the 408 CSF
MCSCF+1+2. These energies are collected in Table 1.
OH

The 211 state of OH was examined by a 2B1 17 configuration
MCSCF function. This was constructed from all spin couplings of a
GVB(2/4) function (correlating the 1: bond and the doubly occupied O

2p1t orbital) and a 2,729 CSF MCSCF+1+2 constructed from all

Y
valence single and double substitutions (of 2B1 symmetry) from the

MCSCF reference space. The total energies are listed in Table 1 while
the dissociation energy (De) is collected in Table 2.
H20

The energy and optimized geometry of H20 was computed with
a 37 CSF MCSCF function (constructed from all spin couplings of a
GVB(2/4) function) and an 18,410 CSF MCSCF+1+2 function derived
from all valence single and double substitutions from the MCSCF
reference space. The equilibrium energies are listed in Table l, and
the dissociation energies in Table 2.
H2 and H

The H (ZS)15 energy was computed with a SCF function using
the (2s,1p) basis. This energy is collected in Table 1. The total
energy of H2 using the (3s,3p) basis was determined with a 3 CSF

MCSCF and 120 CSF MCSCF+1+2 function constructed from all single

and double excitations. The total energies and De are listed in Tables

1 and 2, respectively.

10
W

All ab-initio calculations on the oxygen containing species were
done on a FPS-164 jointly supported by the Michigan State
University Chemistry Department and the Office of the Provost by
using the Argonne National Laboratory collection of Quest-164 codes.
The intergrals were calculated using the program ARGOS written by
Pitzer;l6 the SCF and MCSCF calculations were done using GVB164
written by Bair17 and the UEXP program and related utility codes
written by Shepard.18 The configuration interaction calculations
were performed using the program UCI (and related utility codes)
written by Lischka et al.19

Ab-initio calculations on the sulfur containing species were
done on a Stardent TITAN computer located in the Michigan State
University Chemistry Department using the Argonne National
Laboratory collection of COLUMBUS codes.20

All density and difference density contours were calculated
with the MSUPLOT collection of codes written by Harrison, and all
spectroscopic constants were determined by performing a Dunham

analysis.21

11

Table 1: Total Fragment Energies (au)

Fragment Emin (MCSCF) Emin (MCSCF+1+2)
Sc+ 3D(4s13<11) 45952848 459.52906
8c+ 3B2(3dn13d61) 459.48576 459.49960
8c++ 2D(3dl) 45908187

3 31> 497.50318 49754525
0 31> 44.82254 44.87100
H 28 -0.49928

SH 211 498.11446 -398.l6883
SH2 1A1 -398.73762 498.80756
OH 2H 45.46483 45.52573
0H2 1A1 46.14408 46.21173
H2 12+g(3s/3p) -l.14813 -1.16652

12

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13

W490 AND +5315
A. E N IDE N

In the following discussion only the oxygen-containing species
will be discussed. In all cases, Sulfur may be substituted for Oxygen
with the appropriate change in valence orbital level.

If the two valence electrons on Sc+ form two bonds with the
two unpaired electrons in the ground state of O, the resulting

molecule is a singlet of )3 symmetry and can be represented by the

Lewis structure

+SC:O 12+

where we suppress the explicit representation of the 0 2s electrons.
The ground state oxygen atom may approach the Sc"' in either of two
orientations, according to whether the oxygen 2ps orbital is singly or

doubly occupied.

 

II orientation 2' orientation

14
The local symmetry of O in the first approach is IT and in the
second, 2‘. If 0 is in the TI orientation, 80" must also be locally II and

this may be accomplished using either the ground 483d configuration
(483de or 4s3d1ty) or the low lying 362 configuration (3d03d1tx or

3do3d7ry). These options result in the Lewis structure

 

where one of the 1: bonds results from the singlet coupling of the Sc+
d7r and O 2p1c and the second 1: bond is a dative bond formed from
the lone pair in the 2p7t orbital on O and the empty dz: on Sc+. In the
calculations the It bonds are of course equivalent. The 80"” o electron
is asymptotically either a 45 or 3do. If, however, 0 is in the 21'
orientation, Sc+ must also be locally 2'. This may be achieved using
the (12 configuration, in particular dnx dry. This results in the Lewis

structure

 

0 OR

 

15

where the o electrons are formally from 0. Clearly the equilibrium

structure will be a mixture of the two Lewis structures.

B. W‘“&

The character of both Lewis structures may be incorporated
into a 3 pair Generalized Valence Bond22 (GVB) wavefunction of the

form

‘1’ ~ (core)2 (802-A9oz)(37rx2-v47rx2)(37ry2-v47ty2)

where the core electrons are suppressed for brevity but are always
fully optimized in all calculations. An MCSCF function of this form
which includes all possible spin couplings consists of 37 configuration
state functions (CSFs). A function of this form allows the bonds to
properly separate to the ground state atoms for large internuclear
distances. In particular, the 0 bond (862-1962) separates to the Sc+
4s1 and 0 2p;1 orbitals while the two It bonds together separate to
the Sc+ 3d7r1 and O 2p7t4 configurations. The energy predicted by
this function is shown in Figure 1 as a function of Sc-O separation.
This function predicts an equilibrium separation of 3.095 an and a

dissociation energy, De, of 134 kcal/mol. Also shown in Figure l are

various low lying triplet states. The 323"“ state obtains by triplet

coupling the o bonding electrons in the 12* state. The d8+ symmetry

orbitals were eliminated from the 32* calculation to prevent

collapsing to the 6+ component of the lower energy 3A state.

l6

  
   

 

 

50.0‘ MCSCF Potential Energy / .4
.1
30.0- Sc+ (382, 3d2)+L(3P, p4)
10.0-
0 80‘ (so, 4s‘3d1)+L(3P, p4)
5 '10-0‘ +ScO 32+
> ‘ \
£ ‘30. 0 - \\ \\ 1’
”<5, 50 0‘ fl ’
.. + (a \"x
$00 A
-70.0 -
-90.0-
-1 10.04 ~
T ’l”\ "
-130.0- +Sco ‘2‘“ -

 

 

 

I l I l I l l
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
r(au)

Figure 1. MCSCF potential energies of the 12+, 3A and 32+ states of +ScS and +ScO
relative to the ground state asymptote. Energy is in kcal/mol. The atomic structure at the
asymptote is indicated by both atomic symmetry and valence configuration (L=O and S)

17

One of the triplet coupled electrons becomes localized on Sc+ in an

orbital of 3do symmetry with some 45 character while the
companion electron settles into an oxygen 2pc orbital.

The bond length in this state is longer than in the 121+ state
(3.454 au versus 3.095 au) and the molecule contains two equivalent
7: bonds and no 0 bond. This molecular state should dissociate to the
ground15 3D state of Sc+ and the ground15 3P state of O and its De
relative to this asymptote is 39 kcal/mol. If we imagine forming this
state from the asymptotic ground state fragments we must triplet
couple the spatially extensive Sc+ 4s and the O 2pc electrons and
singlet couple the Sc"’ 3dr: and singly occupied 0 2px orbital. At large
Sc-O separations the 7m: bond would be very weak and the repulsive
triplet coupling in the 0 system dominant. Consequently we
anticipate that this state would be repulsive at large separations and
would have to overcome a barrier to obtain the electronic structure
we see at equilibrium. This equilibrium structure obtains when this
repulsive curve intersects the attractive 32"" curve which separates
to the Sc'i+ (2D; 3do) + O' (2P; 2pc) asymptote. The second triplet is
of 3A symmetry and is obtained from the 323+ by moving the
unpaired o electron on Sc into a 8 orbital. The 8 occupation forces
dissociation to the higher energy asymptote seen in Figures 1 and 2.
Both of these states have 7r,7r bonds and no a bond. Exciting an
unpaired electron from the o orbital on Sc+ to its 8_ orbital puts more
electron density on Sc+ in the 7: region and weakens the 1t,1l: bonds.
This results in the 3A bond length increasing to 3.50 an as compared

to 3.45 an in the 323+ state. This 0 to 6_ excitation also reduces the

repulsion between the unpaired electron on Sc+ and the O 2pc

1 8
electron. That the total energy of the 3A state drops by 15 kcal/mol
relative to the 32+ state suggests the reduced repulsion more than

compensates for the slight reduction in the 7m: bond strength.

The absolute energies (Emin), dissociation energies (De), bond

lengths (re), and vibrational frequencies (we) are collected in Table 3.

CM 8 F R+&§

A MCSCF function that includes the two important

configurations for +ScS is constructed as follows.

‘I’ ~ (core)2 (1002-11l02)(47cx2-v57tx2)(41ty2-v5nyz)

where the structural correlation is achieved with a GVB(3/6)

function22 followed by all spin couplings. This results in 37

configuration state functions (CSFs) of 12+ symmetry under the C2v
point group. The 6 bond (100241102) separates to the Sc+ 4s1 and S
3pz1 orbitals while the two 7: bonds together separate to the 80+
3d1t1 and S 3p7t4 configurations for large internuclear distances

The energy predicted by this function is displayed in Figure 1
as a function of the Sc-S internuclear distance. The De relative to the
ground state products is calculated to be 82 kcal/mol with an
equilibrium separation of 4.048 an.

The 32+ MCSCF is obtained by triplet coupling the 123+ sigma
valence electrons. The Sc 8+ symmetry orbitals were eliminated to

prevent collapse to the 6+ component of the lower energy 3A state.

This results in 25 CSFs. The equilibrium structure has two 7t,7t bonds

19

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20

with one of the triplet coupled sigma electrons localized in a
Scandium 3do+A4s orbital and the other in a Sulfur 3pc orbital.

The bond length increases relative to the 122"" state by 0.267 an
to 4.315 an and the De relative to the ground state asymptote
becomes 18.8 kcal/mol. We expect the long range Sc+ to S interaction

to be attractive (electrostatic) but as the two atoms approach along

the 32‘." curve we anticipate a repulsive interaction between the
triplet coupled sigma electrons (Sc+ 4s + S 3pc) analogous to +ScO.

The 3A function results from moving the Sc+ 3do+A4s electron to a
3d8, orbital. This results in a 25 CSF MCSCF function giving rise to a

7m: doubly bonded species with one electron localized in a Sc+ 3d6_

orbital and the other in a S 3pc. The bond length increases slightly

to 4.377 au while the De relative to the ground state asymptote
becomes 25.6 kcal/mol. The 3A and 32+ energies are displayed as a
function of internuclear distance in Figure l. The De, re and

vibrational frequencies (we) are collected in Table 3.

D. MCSCF+1+2 ESUL'IS FQR +ch

The three states of +ScO described above were studied using
multireference configuration interaction (CI) techniques. For each
state we constructed a CI wavefunction containing all single and

double substitutions from the MCSCF reference space. For example,
the 12"" MCSCF space consisted of 37 CSFs. All single and double

excitations from the space, consistent with the 123+ symmetry, result

in 23,990 CSFs. Several experiments were performed to test the

adequacy of this procedure. In the first we added an additional

21
active 0 orbital to the MCSCF space and generated 81 CSFs. All

singles and doubles from this reference space resulted in 40,996
CSFs. These additional configurations lowered the total energy of the
12+ state by 10 millihartrees (mH) at the MCSCF level and 1 mH at
the MCSCF+1+2 level, but had no appreciable effect on re or Be

In the second we examined the necessity of including the O 28
orbital to the MCSCF (CI) active space. The O 28 orbital was added to
the MCSCF active space (generating 81 CSFs) followed by all valence
single and double substitutions and resulted in 99,463 CSFs. We also
constructed a CI function by allowing all valence single and double
substitutions (including the O 25) from the 37 CSF MCSCF reference
space. This resulted in 76,659 CSFs. The total CI energy dropped by
60 mH for each function while the computed De remained essentially
the same at 145.9 and 144.7 kcal/mol, respectively.

We conclude from these experiments that excitations from the
0 2s orbital are not important in determining the relative energy and

re of the low lying states of +ScO. The size of the triplet states

constructed as all single and double excitations from the MCSCF
reference space was 29,481 CSFs (3A) and 24,133 CSFs (32+). The

+ScO potential curves at the MCSCF+1+2 level are presented in Figure

2 and the calculated re, De and toes are collected in Table 3.

E. W‘Lfl

The three states of +ScS described above were also examined
using multi-reference configuration interaction (MCSCF+1+2)

techniques. The MCSCF+1+2 wavefunction for each state was

22

 

 

 

 

 

 

50.0‘ MCSCF+1+2 Potential Energy ‘
30.0- -
_ 80+ (3132, 3d2)+L(3P, p4) x -
10.0- ‘
0 Sc+(3D,4313d1)+L(3P,p4) /
-10.o - + '-
.. ScO 32+ -
5 40.0- \ -
E
h — \ . ._
8 \\ \M"
é “50.0 " r ‘\ '-
.. + 3 \ _
LLI 800 A ‘\___’
<1 -70.0- -
-90.o~ -
-11o.0~ '-
-130.0- , -
_ +Sc012+ -
-1 50.0 I V I I I l I I
2.5 3. 3.5 4.0 4.5 5.0 5.5 6.0 6.5

r(au)

Figure 2. MCSCF+1 +2 potential energies of the 12+, 3A and 32+ states of +ScS and +ScO
relative to the ground state asymptote. Energy is in kcal/mol. The atomic structure at the
asymptote is indicated by both atomic symmetry and valence configuration (L=O and S)

23

constructed by allowing all single and double excitations from the

MCSCF reference space.
In particular, the triplet states resulted in 34,952 CSFs (3A) and

29,205 CSFs (32+) and the ground state (12+) was 28,337 CSFs. The
predicted energies are displayed as a function of Sc-S distance in
Figure 2. The inclusion of dynamic correlation accounted for by the
MCSCF+1+2 function drops the total energy by around 60 mH but has
only a small effect on the computed dissociation energies and no

effect on the state orderings. The De, re and we are collected in Table

3.

ELEQBQNIQ DISTRIBLJflON
A. +309

Included in Table 4 are the valence orbital populations23
predicted by the MCSCF function for various states of +ScO. Note that
in the 12” state there is very little Sc"' 48 character and the Sc+ ion
has lost electrons to neutral 0. The charge distribution may be
rationalized by imagining the in-situ Sc+ ion in the d71:,‘d7ty
configuration interacting with the O atom in the 2p<3227rx27ry
configuration. Oxygen first donates charge to the empty 3do on Sc+
via the dative interaction of the doubly occupied 0 2p. As charge

leaves 0 in the 6 system it returns in the It system.

24

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While the total charge on the Sc+ ion in the 32"“ state is similar
to that in the 12"" state (+1.40 vs +1.28) the character of the
electrons is very different. In particular, in the 32"’ state there is a
large 45 component and a significantly reduced 3dr: occupation. We
can rationalize this by noting that the 32‘." may be formed from the
123+ by triplet coupling the o bonding electrons. This localizes one
electron in o orbitals on Sc and the other in a 2pc on 0. As a result of
this transfer the oxygen atom becomes more positive and attracts
electrons into the 2px orbitals, considerably reducing the Sc+ 3dr:
occupation. The choice Sc+ has to make is the relative amount of 4s
and 3do character to allot to its unpaired electron. If the in-situ
character of Sc was Sc++ we would expect the unpaired electron to be
primarily 3do.15 The observed 40% 4s, 60% 3do reflects the
intermediacy of the Sc+ charge (greater than +1 but less than +2).
The electron distribution in the 3A can be understood by noting that
the 3A is formed from the 323"" by exciting the unpaired o electron to

a d orbital, precluding any 4s character.

26
B. his

The valence orbital populations23 as predicted by the MCSCF
functions for the +ScS states are collected in Table 5. We find that
the Sc+ 4s contribution to the 121+ state is small (0.17) though non-
negligible and that Scandium has lost some electron density to S. The
equilibrium structure is described as Sc+ in a 3F (3d1tx,3d1ty) state in-
situ bonded to S with two 1c,1t bonds and a (primarily) S 3pc to Sc+
3do dative bond. This structure arises when S donates charge in the
sigma system (sigma dative bond) followed by It density being
transferred back from Sc+ to S. The distribution of bonding electrons
in the +ScS 3A and 32* states are similar with both markedly
different from the 122+ state. We find that precluding the sigma
dative bond formation results in significantly more 1!: density
(approx. 0.30e per bond) being transferred from Sc+ to S. When the
sigma electrons are triplet coupled one becomes localized on
Scandium and the other on Sulfur. This results in relatively less
sigma electron density on S in the 3pc orbital, causing a greater
propensity to attract density into its 3px orbitals thus reducing the
Sc 3d1I: occupation.

The contribution of the 80" 4s to the 32"“ state is twice that for
the 122+ state. This can be understood by realizing the 32‘." state
results from triplet coupling the sigma electrons in the 12"“ with the

localized electron on Sc+ becoming of 3do+k4s character.

27

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This structure strikes a balance between the lower energy
Scandium 4s3d configuration and the 3d2 configuration which
distorts the charge perpendicular to the sigma orbitals, effectively
draining charge from the internuclear region.

The 3A state arises from moving the Sc 3do+h4s electron to a
Sc+ 3d8_ orbital. The bonding structure doesn't change, indicating
that both electrons are simply spectators. The MCSCF gross atomic

charges for +ScS are collected in Table 6.
C. CQNIPARISQN QF +$Q AND +$§

In comparison to ““800 we find that in general the +ScS gross
charge transferred is less as expected from the longer bond distance.
We find the sigma bond composition in +ScO 12+ to be an O 2pc and
Sc+ hybrid composed of the 43, 4pc and 3do orbitalsl. Similarly, the
+ScS 12+ sigma bond is found to be a S 3pc plus Sc"‘ 45, 4pc, 3do
hybrid with, however, the 4pc and 4s contributions inverted relative
to +Sc0. In +ScS 12‘." the Sc+ 4s orbital is much more important than
the 4pc. This larger 45 contribution weakens the sigma bond
because of the lesser overlap with the S 3pc. This greater
importance is caused by the larger internuclear separation and the
consequently smaller in-situ gross charge on Sc in +ScS. This causes a
larger portion of the wavefunction to be composed of the 4s3d
configuration relative to +800. This smaller overlap would result in a
smaller charge donation to Scandium in the sigma system and hence

a smaller amount of back donation in the 1: system as suggested in

Table 5.

29

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30
BMW

The electron distribution in the 122+ and 32+ states is so

different that it is easily seen at the total density level. The total

density (p) is obtained from the MCSCF N08 with the following

equafion:

p(R) = < vl ism-R) I v >.
i

where ri denotes the electron coordinates and R is the field

coordinate. The total density simplifies to

mm = 20i¢i2(R)

O

1
where o is a spatial natural orbital (NO) and n1 is the occupation of
the orbital. The difference density contours (DDCs) are obtained by
subtracting one density from another.

Figure 3 shows the total electron density of +Sc0 contoured in a
plane containing both nuclei for these two states. The Sc atom is to
the left of 0. Positive contour levels are indicated by solid lines and
negative contours by dashes. For both states, there is an decrease of

density in the 0 space relative to the ground state. This is caused by

the triplet coupling of the valence sigma electrons. Dashed lines in

the It space reflect the concomitant increase in 1: density.

31

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+Sc0( 12*) Total Density
8“” l l l
' F
.2
at .
>
at '-
-8.0C I
4.00 z-axls 8.00
+ 3 + . + 3 .
Sc0( 2 )Total Densnty Sc0( A )Total ”CBS!”
8.-- l l L 8.00 l l j
.2 .2
g . . x . .
s 1'
i I I L
-8.“ . ace 1
4.00 z-axls 8.00 4.00 z-axls 8.100
1 + 3 + 1 3
*Scm 2 - 2 )Diff. Density +Sc0( 73- A )Diff'. Density
8 -- l I 1 8°C 1 ' '
I ll 7;: . I '
.2 .2
X . I a I P
1 s
-8.00 I 8.00 I
4.00 z-ax|3 8.00 4.00 z-axls 8.00

 

 

Figure 3. MCSCF Total density contours (TDCs) and difference density contours (DDCs) for the 122+, 3A and 32+ states

of +SocO. The DDCs are molecular differences where the indicated triplet state is subtracted from the ground 12+ state. The
triplets are at the equilibrium geometry. The contour levels range from 0.0025e to 1.289 (T DCs) and -0.04 to +0.04 (DDCs).
Each level differs by a factor of 2. No zero contour is displayed and negative contours are indicated by a dashed line.

32

AB- 1 ATI N F+M_LND +5231!
A.+§9Sll.

The hydroxide can be formed by adding a H atom to +ScO. Both

the 3A and 32* states have an unpaired 2p electron on oxygen and
singlet coupling the H ls electron to the oxygen electron results in
the linear 2A and 223+ states of +ScOH. These two states have the
Lewis structure

(1r / o

,+Sc=O—H
2 in
d5 or do + K4

in which the unpaired electron is localized on Sc+ in either a d8 (2A)
or sigma orbital of mixed do and 45 character (22+). MCSCF functions
which correlate the three bonds and allow all spin couplings consist
of 76 CSFs in Cs symmetry for each molecular state. We optimized
the +Sc-O and OH bond lengths as well as the +Sc-O-H angle at the
MCSCF and MCSCF+1+2 levels. Both electronic states are linear and
the total energy, bond lengths and various dissociation energies are
collected in Table 7. The electron populations in the valence orbitals

and the charges on each atom are collected in Tables 4 and 6.

The 2A state of +ScOH is calculated to be 17.2 kcal/mol lower

than the 221+ state, which is very similar to the corresponding 3A-32+

33

separation of 19.9 kcal/mol for +ScO (both calculated at the
MCSCF+1+2 level of theory). As we see from Table 4, the H atom has
little effect on the charge distribution on Sc+ and in particular on the
character of the unpaired electron.

Figure 4 shows the electron density in the 221+ state of +ScOH
minus the density in the 322+ and 122+ states of +ScO. The +ScOH (22+)
- +ScO(1£+) difference density illustrates the negligable effect
bonding of H to O has on the character of the Sc+ non bonding
electron. The difference density is very similar to that of +ScO(1£+)-
+ScO(32‘.+) (Figure 3) and results from a similar mixing of do and 4s
orbitals. The +ScOH(221+)-"'ScO(3).“."') density shows little difference in
the Scandium structure, in-situ, and indicates that slightly more +Sc-
O sigma density is present in the hydroxide.

The relative energies of +ScO and +ScOH are shown in Figure 5.
Experimental values (corrected for zero point energy) are shown in
parenthesis. Our calculated bond energy for free OH is 97.6 kcal/mol,
approximately 10% lower than the experimental De (corrected to OK)
of 106.8 kcal/mol.24 We have two options for the OH bond strength
in +ScOH(2A). First, we may break the OH bond along the A potential

C 11 IV C
+ScOH(2A) => +ScO(3A) + H(28)

which requires 139.1 kcal/mol, significantly larger than the free OH
value. This enhanced OH bond strength obtains because the +Sc-O
and O-H bonds are strongly coupled in +ScOH(2A).

34

+ScOH( 22”) Total Density at equilibrium

mo 1

 

 

 

 

 

>.
- L
w I fir I
4.00 z-axis 12.00
+ 3 . + 3 + .
ScO( A )Total Dens1ty ScO( 2‘. )Total Densrty
son 1 l l m l 1 1

 

 

 

 

 

z-axis 110°

“Scout 22*) -+SCO( 3A ) Diff. Density
m 1 l 1

 

 

 

.em

 

 

4N

T

l
z-axis

l

12.00

 

 

 

 

 

 

 

 

 

4.0.1 I z.a'xis T 12.00
+ScOH( 22*) -+ScO( 32*) Diff. Density
m l 1 1
a
'9 .. I” i—
6 K,
ti.
'8'“, T l r
400 z-axis 12.00

Figure 4. MCSCF total density contours (TDCs) and difference density contours (0003) for the 213+ state of +ScOH and the
1}? and 32* states of +ScO The 0003 are molecular differences where the +ScO states are subtracted from the +ScOH

density. The +ScO states are at the equilibrium *ScOH (Sc-0) geometry. Contour levels range from 0.00250 to 1.288 (T003)
and 0040 to 0.040 (0008). Each level differs by a factor of 2. No zero contours are displayed and negative contours are indicated

by dashes.

35
When H bonds to the unpaired 2pc electron in +ScO(3A) the 0 2s and

2pc hybridize, simultaneously strengthening the OH bond and
forming a dative bond in the empty 0 space of Sc+ using the
companion to the O-H bond hybrid. This symbiosis also manifests
itself in a stronger than expected Sc-O bond strength in +ScOH. From

Figure 5 we see that the +Sc-O bond strength in +ScOH is 108

kcal/mol, intermediate between the strength of a +ScO(2A) double
bond (66.4 kcal/mol) and the +Sc0(12+) triple bond (146.0 kcal/mol).
Our computed +Sc-OH bond strength (108 kcal/mol) is significantly
higher than that reported5 by Magnera et al. (87.8 kcal/mol). These

experiments determine the +Sc-OH energy from the parent molecule,

(HzOScOH)+. If the structure of this species where of the form
HzO---+Sc-OH, then the intact +Sc-OH would prefer to be in its ground
2A state in-situ. The 2A state positions the Sc non-bonding electron
in a 3d8- orbital perpendicular to the internuclear axis and would
minimize the repulsions to an electrostatically bound H20 molecule.
The energy of the 222+ state of +ScOH is 17 kcal/mol (MCSCF+1+2,
Table 7) higher than the 2A. While an H20 molecule should still be
able to electrostatically bind to the Sc atom, their should be more
repulsion between H20 and the now in-axis 3do electron. This would
likely increase the 223+ <— 2A separation and could possibly account
for the 20 kcal/mol discrepancy.

The second option for the OH bond breakage is the

thermodynamically lowest path

+Sc0H(2A) => +Se0(12+) + H(2S)

36

 

 

 

 

 

 

 

 

 

 

 

205.6
— l l _
200 A +Sc(3o)+0(3P)+H(23)
b
159-‘ 97.6(106.8)
150— +Sc0(3z+l+H(28) / O-H Bond
139.1
{-5 +ScO(3A)+H(28) l
E .
'53 108.0 (88)
5,, 100— \ 91.9 +Sc(30)+0H(2n) ‘—
> a H-+Sc0(22+)
9 146(159: 7)
Q)
c
LU
V 59.6
50"“ +Sc0(12+)+H(23) .—
17.2
+Sc0H(22+)
0— i 0.0 __
+Sc0H(2A)

 

 

 

Floors 5. MCSCf+1+2 relative energies (kcal/mol) (nunbers in parenthesis are experimental values)
8 Ref (3), b Ref (24) Value given as De corrected to UK, c Ref (5).

37

which requires 59.6 kcal/mol. This is much lower than the free OH

bond strength, reflecting the differentially stronger +Sc-O bond in
+ScO(12+) compared to +ScOH(2A).

B.+&Sfl

+ScSH states of 2A and 22"" symmetry can be formed when a H atom
bonds to the S 3pc electron in +ScS 3A and 32“”, respectively. This

gives rise to the Lewis structures

(2A) 3d6./\. +5 egg—H

87:7

1C

and

m

2+) 3do +x4s - +Sc <—S—H

TC.

(2

The 2A MCSCF wavefunction was constructed to correlate the 3
bonds in a GVB way, followed by all spin couplings, and results in 76
CSFs. The 22*" MCSCF was constructed to correlate the two 1: bonds in
a GVB way and three valence sigma orbitals. All spin couplings on

this function result in 144 CSFs. All calculations were performed

under the C2v point group. The MCSCF+1+2 wavefunctions were

38

constructed by allowing all valence single and double substitutions
from the MCSCF functions and results in 138,529 CSFs for the 2A and
149,312 CSFs for the 22"”. The Sc+ 3d8+ orbitals were eliminated

from all +ScSH 22“” calculations to prevent collapsing to the lower

energy 2A(8_,_) state. The total energy, optimized geometries and
bond dissociation energies are collected in Table 7.

The 2A state is calculated to be 14.2 kcal/mol (MCSCF+1+2)
lower in energy than the 22+. This is similar to the (MCSCF+1+2) +Scs
3A 32+ difference of 12.6 kcal/mol and slightly smaller than the
corresponding +ScOH difference of 17.2. 1

The MCSCF electron distributions for the +ScSH 22"” and 2A
states are collected in Tables 5 and 6. We find the bonding of H to
+ScS has little effect on the +Sc-S 1r bonding structure. Analysis of the
natural orbitals (N05) and populations does, however, reveal
significant changes in the sigma structure. In both states the S 3s
and 3pc orbitals are hybridized, one hybrid bonding to H and the
other interacting with the Sc+ 4s + 3do orbitals.

The strength of the 80+ to SH sigma dative interaction can be
estimated by examining the bond energies from Tables 2, 3 and 7.
Our calculated SH (211) bond strength (D0) of 77.0 kcal/mol (De=78.0
kcal/mol) is approximately 6% lower than the experimental value of
81.4 kcal/mol.24
If the +ScSH (2A) S-H bond is broken along the A potential curve

+ScSH(2A) => +ScS(3A)+H(23)

39

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58.8.8.3

3.825..

40
we obtain an MCSCF+1+2 AE of 94.4 kcal/mol, 16.4 kcal/mol greater
than the free (MCSCF+1+2) SH bond strength. Dissociating +ScSH
along the path

+ScSH(2A) => Sc+(3D) + SH(2r1)

requires 56.6 kcal/mol and is, as required, 16.5 kcal/mol larger than

the +ScS(3A) bond strength of 40.1 kcal/mol. These enhanced bond
strengths are the result of bond formation to H. As the S to H bond is
formed, using the S SS and 3pc orbitals, the companion 3s :1: 3pc
hybrid orbital interacts with Scandium causing a simultaneous
strengthening of the Sc"’ to SH bond. This is half that found for +ScOH
where a stabilization energy of 43 kcal/mol was observed

(MCSCF+1+2).1
+ScSH(2A) can also dissociate along the adiabatic pathway
+ScSH(2A) => +3cs(lz+)+H(23)

and requires 37.6 kcal/mol. This is significantly less than the free SH
dissociation energy and reflects the differentially stronger bond in
+8cs(12+) relative to +ScSH(2A).

The gross atomic charges are collected in Table 6 and indicate
that bonding H to +ScS does not change the charge on Sc but increases
the anionic character on the Sulfur. The increase in S charge relative

to the +ScS molecule is approximately 0.2 electrons.

41
C-CQ___M_MPARI N F+§§N_H, “‘ScflflAED ”@283

The species +ScNH,6 +ScOH and +ScSH can all be described with

the Lewis structure
+ —
50:14-11

where if L = O or S both the o and one I bond are dative bonds while
for N only the 0 bond is formally dative. The bonding of H to L

causes the ligand's valence s and po orbitals to mix as suggested

below.

Sc L H

< H to L
a Empty \ Lto Sc
45+)» 3 d o Valence Datwe 30nd Bond

8 orbital Valence
poorbital

    

For L = N and 0 this results in a sigma dative bond that stabilizes the
Sc-L interaction by 43 kcal/mol. Since the S atom is larger than
either 0 or N the bond length is longer than in either +ScO or +ScN.
Moreover, the larger size also decreases the amount of stabilization
afforded by formation of the sigma dative bond to 16.6 kcal/mol.
The increased Sc to S bond length also affects the detailed structure
of the spectator electron density. In +ScOH the Scandium spectator

electron is composed of (in order of decreasing importance) the Sc+

42

3do,4s,4pz with the 4s contributing up to 44% of the charge density
(22‘.+ state). In +ScSH, the mixing also goes as Sc+ 3do, 4s, 4pz, but
with the 4s only contributing around 23% of the charge density (22"'
state). This is indicative of the amount of Scandium 4s available for
the sigma dative bond. In particular, the 4s contribution to the
dative bond is greater in +ScSH than in +ScOH. The structure of the
dative bond and the similarity in the +ScOH and +ScSH structures is
illustrated in the MCSCF difference density contours displayed in
Figures 4 and 6. In both Figures the 221+ +Sc-SH (+Sc-OH) total
density at a geometry near equilibrium has the fragment +ScS(+ScO)
triplet state density subtracted from it. The +Sc-L molecule is
maintained at the +ScLH geometry. This clearly indicates which
orbitals are used in constructing both the sigma dative bond and the
1: bonds. Specifically, the +ScSH(2A)-"'ScS(3A) difference density
(Figure 6) shows a much increased density in the Sc-S sigma system
relative to +ScS(3A). The MCSCF+1+2 bond dissociation energies for
several states of +ScSH, +ScOH and +ScNH have been collected in Table
8.

A NPD

A. _H_-+S_¢_Q

If we bond to the unpaired electron on Se in the 3A or 32‘.”
states of +ScO we form H-+Sc0(22+). The Sc-H bond strength, relative
to the 3A state of +ScO is calculated to be 47.2 kcal/mol (MCSCF+1+2

level), a typical25 Sc-H bond strength.

43

+ScSH( 22+) Total Density at equilibrium

1.00 L I 1

 

 

 

 

 

.gq L
$.
w I I I
490 z.axis 12.00
+ 3 . + 3 + .
ScS( A )Total Densnty ScS( 2‘. )Total Densrty
son 1 L L m L 1 1

 

 

   

 

 

 

 

 

 

 

 

m I I I .m I I I
400 z-axis 110° 41» z-axis 110°
+Seem 2L”) -+ScS( 3 A ) Diff. Density *Scsm 22+) - +Ses ( 32”) Diff. Density
m 1 1 1 1 m 1 1 1
l
L « i-

   

 

 

 

 

 

w r 4.00 t t t
41” z-axis 110° 40° z-axis 11‘”
Figure 6. ucscr total density contours (T003) and difference density contours (DDCs) tor the 22+ state of +SesH and the

12* and 32* states of +8.6 The 0003 are molecular differences where the +806 states are subtracted from the +36H

density. The +ScS states are at the equilibrium +S$H (Sc-S) geometry. Contour levels range from 0.0025e to 1.28e (T DCs)
and -0.04e to 0.049 (DDCs). Each level differs by a factor of 2. No zero contours are displayed and negative contours are indicated

by dashes.

 

44

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There are three isomers with this empirical formula: the two
electrostatic complexes,
Sc+m 0H2 (triplet)
and
H2---+ScO (singlet)
and the insertion product

H-+Sc-OH (singlet).

The electrostatic complex involving intact H20 was studied by
Rosi and Bauschlicher.7 The Sc+m 0H2 complex is bound, relative to
the ground state products, by 36.2 kcal/mol with a Sc+ to 0H2
distance of 4.296 au. The H20 was constrained to the SCF geometry.

We will focus on the two remaining isomers. Consider first the
electrostatic complex involving intact H2. It is easily seen that this
will be an exoergic product of the reaction of Sc+ with H20. It
requires 219 kcal/mol (AE) to dissociate H20 into its atoms24 and we

regain 103 kcal/mol (AE) when H2 is formed24 and 159i7 kcal/mol

when +ScO(12+) is formed3. AE for the reaction
Sc+(3D) + H20(1A1) = +Sc0(12+) + H2(12+g)

is at least 36 kcal/mol exothermic. Detailed calculation at the MCSCF
and GVB+1+2 levels result in the energies collected in Table 9. Our

explicitly calculated AE for the above reaction is 32.6 kcal/mol at the

GVB+1+2 level.

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47

The electrostatic complex is bound by an additional 2.5 kcal/mol,

making our calculated AE=-35.1 kcal/mol for the reaction

Sc"’(3D) + H20(1A1) = H2---+Sc0( 1A1)

The insertion product may be formed from either the 2A or Z)?"
states of +ScOH by coupling the second H atom to the unpaired
electron on 86". The resulting molecule has 4 formal electron pairs (a
+Sc-l-I, O-H and two +Sc-O bonding pairs), and an MCSCF function
which correlates each (in the left-right GVB sense) and includes all
spin couplings consists of 150 CSFs. The bond lengths and bond
angles for the planar structure were optimized and the results are
shown in Table 9. Also listed is the optimal geometry and associated
energy obtained from a CI wave function which includes all single
and double excitation relative to an 8 configuration (4 pair) GVB
function (which generates 112,088 CSFs of 1A symmetry). The single
particle basis for this CI were the natural orbitals from the MCSCF
function. This calculation places the insertion product ~5 kcal/mol
lower than the electrostatic complex.

The errors in these calculations increase in the order

Sc+---OH2 < H2---+Sc0 < H—+Sc-OH

and improved calculations should favor the insertion product, suggesting
that it is the global ground state. The +Sc-H bond length is 3.50 an and

the bond energy is calculated to be 50 kcal/mol

4 8
H-+Sc-OH => H(ZS) + +sc-0H(2A) AB = 50.5 kcal/mol

which is remarkably similar to the 3.52 an and 50.7 kcal/mol
calculated by Alvarado-Swaisgood and Harrison25 for +ScH(2A).
The computed AE for removing the O-H hydrogen

H-+Sc-OH => H—+Sc-O + H AB = 141 kcal/mol

is 141 kcal/mol, virtually the same as that computed for
+3coH(2A) = +Sc0(3A) + 11(23) AE = 139 kcal/mol

C. QQMPARIsQN MTH THE §§+ + NH3 sYsTEM

It is interesting to compare these results with those reported
recently for the Sc+ + NH3 system.6 The ground state of +ScN is of

22+ symmetry and has a calculated bond energy (De) of 63.1
kcal/mol. The molecule has two It bonds and no 6 bond. Its Lewis

structure is

[N f N

+Sc_

K.

When the N atom bonds to an H atom, its 25 and 2po orbitals

hybridize - one component reaching out to bond the H atom while

49

the companion component forms a dative bond in the empty valence

0 space of Sc+.

113

0
MM.
K.

The resulting +Sc-NH bond is calculated to be 106 kcal/mol, some 43
kcal/mol stronger than the Sc-N bond in +ScN.
The ground state of +Sc0(1)3+) has a triple bond

1!:
+82!

K

1!

O’

 

with no unpaired electron on 0. However, the 3A state is a 1m: state,

similar to +ScN, except that it has been formed formally from a

dative interaction in the 1: system

 

50

where we show both 1: bonds as being equivalent, of course. When

the O atom is approached by an H, it will also hybridize its 2s and

2pc orbitals forming a covalent bond to H and a second dative bond

(in the a system) to Sc+.

aMfScEOH 0R ,+SCEOH

The +Sc-O bond strength in this molecule is calculated to be 109
kcal/mol, 43 kcal/mol higher than in +ScO(3A) and essentially the
same as the Sc-N bond strength in +ScNH. It is interesting that the o
dative interaction has stabilized both the +Sc-O and +Sc-N bonds to
the same extent, 43 kcal/mol.

This suggests that the +M-L bond energies in the pairs

5'? 54+“

’"rEOH (32') and . +TENH (2A)

5+

and

o—»: +V:OH (42-) and . +VENH (32-)
a, a,

51

will be similar. Indeed, Armentrout et al.26 have determined D0 for

the +V-L pairs and finds 100 kcal/mol for +V-OH and 102 kcal/mol
for +V-NH. Since these two bond strengths are similar, the unpaired
o electron in +VOH must not interfere with the o dative bond. It
would be very interesting to know the detailed atomic orbital

composition of this electron. Finally, The +Ti-L bond strengths were

previously found to be essentially the same with +TiOH Do=ll3
kcal/mol and +TiNH Do=lll kcal/mol.5927

D. 352+;Sflz

The MCSCF wavefunctions for the 3A1 and 3A2 states were

constructed under the Cs point group by pairing, in a GVB way, the
two S-H bonds and in-out correlating the out of plane S 3p1t2 orbital.
The Scandium 4s and 3d orbitals were maintained singly occupied.
All spin couplings from this GVB function result in a 126 CSF MCSCF.
The MCSCF+1+2 functions were constructed from all valence single
and double substitutions from the 8-configuration GVB function (3

GVB pairs and 2 singly occupied orbitals), using the optimized MCSCF
NOs as the orbital basis. This results in 216,530 CSFs for the 3A2

state and 221,182 CSFs for the 3A1. A MCSCF+1+2 function
constructed from the full MCSCF reference space was not possible. It
was found, however, for the Sc+ + 0H2 studies1 that the GVB
correlation plus all valence single and double substitutions using the
MCSCF NO basis accounts for almost all of the energy. In particular,
the MCSCF+1+2 functions for +ScOH using both the MCSCF and GVB

basis results in a total energy difference of only 2 kcal/mol.28 In all

52

calculations the SH2 geometry was constrained to that optimized

with a 37 CSF MCSCF function and a planar geometry was selected.

This technique was previously shown to be adequate for the

electrostatic +Sc-OH2 systems.7

The 3A2 and 3A1 states differ only in location of the 3d
electron on Scandium. In the 3A2, the electron occupies a 3d8,
orbital while in the 3A1 it occupies a 3d8+ orbital. In both states the
companion electron is in a 45 orbital. This subtle difference results
in nearly degenerate states with the 3A2 only 1.8 kcal/mol lower in
energy than the 3A1. The near degeneracy arises from the large +Sc
to SH2 distance in the molecule.

We find the 3A2 MCSCF+1+2 interaction energy to be 11.39
kcal/mol and the 3A1 to be 11.38 kcal/mol. The optimized 3A2 +Sc-
SH2 distance becomes 5.454 au while the 3A1 distance is the same at
5.456 an. The optimized geometries and total energies are collected
in Table 10 as are the corresponding +ScOH2 values.

The interaction energy of the electrostatic species Sc+SH2 arises

primarily from the charge-dipole term in the energy. The

experimental dipole moment of SH2 is 0.97 D24 while that of H20 is

1.85 D.24 The simple charge-dipole energy expression is E=qtt/R2
with u the dipole moment of SH2, q=1 and R is the distance from Sc+
to the center of charge on SH2. Using this expression the interaction

energy of Sc+ + SH2 should go as

E(Sc+SH2) a E(Sc+OH2)*(uSH2/uOH2)*(RSc-0H2/RSc-SH2)2.

53

Table 10: 3A2 and 3A1 states of Sc+SH2. Optimized Geometries, Total Energies.

4: (rs'"
K H
3A2 J®mn < >13
ads- ) c H
I'Sc-s 1’s.“
MCSCF MCSCF+1+2

Energy(au) - 1 15828051 - 1 158.3531

5.608 5.459

rsc_s(au)

4: rs'“
3 k ( H
A1 569"" < 3.
sea, ) (H
l'Sc.s rS-H
MCSCF MCSCF+1+2
l Energy(au) 415828050 -1158.35310
rsc_s(au) 5.609 5.456

 

For all calculations the SH2 geometry was constrained to B(deg)=90

and rS_H(au)=2.60.

 

Sc+SH2 =>

Energetics (CI)
3A2 state

Sc+ + SH2. AB: 11.4 kcal/mol

Sc+SH2 => Sc+ + S + 2H. AB: 176.9 kcal/mol

Sc+SH2 =>

3 A 1 state

Sc+ + SH2. AB: 11.4 kcal/mol

Sc+SH2 => Sc+ + S + 2H. AB: 176.9 kcal/mol

54
This simple expression suggests that E(Sc+SH2)s E(Sc+OH2)*0.33.
Using the E(Sc+OH2) value3 of 36.9 kcal/mol yields E(Sc+SH2) = 12

kcal/mol, only 5% larger than our determined MCSCF+1+2 value of
11.39 kcal/mol.

E. 3255+

This molecule is characterized as intact H2 electrostatically
bound to ground state +ScS(IZ‘.+). A MCSCF function that is composed
of 4 GVB pairs describing the 4 bonds in the molecule plus all spin
couplings results in 74 CSFs under C2v symmetry. The MCSCF+1+2
function was constructed as all valence single and double
substitutions from a 16 CSF GVB function using the optimized MCSCF
NOs as the basis. This results in 107,832 CSFs. The H-H and H2-+ScS
distances were optimized with the Sc-S distance constrained to 4.00
au which is close to the minimum energy (between 3.9 and 4.1 au).
The optimized geometries and total energies are collected in Table 8.
The MCSCF+1+2 interaction energy is determined to be 3.5 kcal/mol
while the reaction product, HZ-Sc8+, is exoergic, relative to Sc++SH2,
by 34.5 kcal/mol.

The formation of H2°--+ScS from Sc+ + SH2 requires dissociation
of SH2 (165 kcal/mol MCSCF+1+2, Table 2), followed by formation of
H2 and +ScS(12:"’) with a small contribution from the electrostatic
interaction. Formation of +ScS(123‘*') recovers 97 kcal/mol
(MCSCF+1+2, Table 3) while the H2 De was determined to be 105
kcal/mol (MCSCF+1+2, Table 2). This suggests that the reaction
energy should be around 37 kcal/mol. Detailed MCSCF+1+2

55

calculations result in an energy of 34.5 kcal/mol with the
electrostatic interaction accounting for 3.5 kcal/mol. The computed
interaction energy of 3.5 kcal/mol is similar to that found for
H2---+Sc0 (2.50 kcal/mol, Table 8) and also that for the +Cr-"H2
system (3.58 kcal/mol).14 The optimized geometry and energetics

are collected in Table 11.

F. .H.+$_SH

The H+ScOH molecule was determined to be the ground state

reaction product of Sc+ + H20. Using simple bond additivity
arguments and a De(+Sc-H) of 50 kcal/mol (a typical +Sc-H bond

strength)25 suggests that H+ScOH is exoergic by only 1-2 kcal/mol or
even isoergic with the reaction products. Including the dative bond
stabilization energy of 43 kcal/mol, however, drives this product to
the final reaction ground state. By analogy, the H+ScSH product
without stabilization would be exoergic relative to the products by
around 3 kcal/mol. Inclusion of the 16.6 kcal/mol stabilization
energy does indeed lower the total energy but not enough to
compete for the ground state. Furthermore, this structure does not
seem to be a minimum on the reaction energy surface. The linear
structure is a saddle point on the molecular surface with a MCSCF
energy, relative to Sc+ + HZS, of 4 kcal/mol. Bending the two
hydrogens to either the trans or cis conformation results in a marked
decrease in energy. In particular, the structure with both hydrogens

90 degrees from the internuclear line and in the cis conformation
drives the MCSCF energy to around 16 kcal/mol below Sc+ 4» H28.

56

Table 11: H2°"+ScS Optimized geometry, Total Energy and Dissociation Energies.

H
Filllll+sc S
H
rH-H

 

 

I.IIIScS
MCSCF MCSCF+1+2
Energy(au) -1158.31501 -1158.38999
rH-H(au) 1.462 1.444
rH2_SCS(au) 5.210 5.063

rSc_S(au) was constrained to 4.0 an for all calculations.

Dissociation Energies (CI)
H2-+ScS 2) H2 + +ScS. AE= 3.6 kcal/mol

H2-+ScS => Sc+ + S + 2H. AE= 178.7 kcal/mol
H2-+ScS => +Sc + SH2. AE= 34.5 chmol

57
This lies below the optimized Sc+SH2 product. Our results

indicate the transition from H-+ScSH to H2---+ScS is an energetically

favorable process with no apparent barrier. This process is
illustrated in Figure 7, where the minima correspond to the
optimized ground state products and the saddle point refers to the

insertion product.

MI Y-PREDI N

Recent theoretical studies of Sc+ + H20 suggests the ground

state reaction product is the insertion product, H+ScOH , and while
the experimental work of Magnera et a1.5 didn't rule out that
possibility, it was suggested that the product was the electrostatic
+Sc-"OH2 species. The discrepancy between the expected
experimental and the theoretical result is caused by a breakdown of
bond additivities resulting from the induced sigma dative bond in
+ScOH stabilizing the system by 43 kcal/mol. In the Sc+ + SH2
reaction we find the induced sigma bond stabilization energy to be
only 16.6 kcal/mol, with the consequence of drastically changing the

relative energies of the reaction products. Therefore, while in the Sc+

+ H20 reaction the products order in total energy as
H+ScOH < Sc+OH2 < H2---+Sc0
in Sc+ + SH2 they order as

H2---+Scs < H+ScSH < Sc+SH2

58

+Sc + SH2 Asymptote

 

     

< 11.39 keel/mol
> 16 keel/moi + 1 a
SC°“SH2( A2)
345 keel/moi +Sc°°° SH2( 3A2 )

  

Reaction Energy

 

Hy’Scm 1A1)

 

 

q (generalized coordinate)

Figure 7. Illustration of the energetics amongst the $0“ + SH2 reaction products.
Minima on the curve are MCSCF+1 +2 optimized values. q is a generalized
coordinate.

aEnergy Expected from the Sc+ 1D c 3D separation of approximately 7 kcanol (ref (15)).

59

These results are collected into Figures 8 and 9, which depict
the relative energies of the studied products; the energetics are also
summarized in Table 12.

In summary, the ground state reaction products of Sc+ with

SH2 and 0H2 can be understood using three physical properties.

1) Dipole moment of the ligand.

2) ground state +ScL De (L is the ligand atom that bonds to Sc").

3) The stabilization obtained when H bonds to +ScL to form +Sc-L--H.
These estimates are sufficient to predict the order of the ground

state reaction products. This analysis is expected to be applicable to

ligands such as SeHz.

N IN

1. The ground state of +ScO is of 12+ symmetry. Our computed Do

of 144.4 kcal/mol compares favorably with the experimental value

of 159:1:7 kcal/mol.

2. A major factor contributing to the strength of the +Sc-O bond is

the dative bond formed between the 2pc electron pair on O and the
empty 0 valence orbitals on Sc+. This suggests that the ground state
of +Ti0 will be of 2A symmetry (° +TiEO) while +VO will be 32'
(:+VEO ) (as observed).29 When the metal valence o orbitals are
not empty the o dative structure will compete with a structure
having a singlet coupled oxygen-metal 0 bond and a dative bond in

the 1r system. In +CrO for example, the o dative structure

200.0

150.0

Energy (kcal/mol)
8
¢

50.0

0.0

60

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+
— I
l——

46.5
_. l’ Sc 0(3):) + 2H
— 66.4
‘ +3ScO( A) + 211
214.7
— +Sc + 0H+H
_. 146.0
— 108.0
—— +Sc0(l}.‘.+) + 211 i
117.1
188.4
+Sc 0H(22+) + H v
”" + 2 In 250.9
__ ScOH( A) + H
+ /

 

 

 

 

_ amok :SCO\‘+ scmonza V

 

 

Figure 8. MCSCF+1+2 relative energies of the Sc+ + SH2 reaction.

3 ref (7).

 

Energy (kcal/mol)

61

+Sc + SH2—>Products.

 

 

 

 

 

 

 

 

 

 

 

 

 

+Sc + S + 2H
_ I .
‘ 27.6 '7
150.0 40 1
. +
r- | *Sc3132 ) +211
_ 0 +ScS(3A) +211
165-5 96.9
100.0
— +Sc + SH+H
120.4
_ A
__ +ScS(12+) + 211 v
_ 56.6
2
50.0 —- 875 *Scsm 2”) + H v
*- 2
+Scsm A) + H E”
_ 176.9

 

 

 

 

 

 

 

 

 

 

 

 

Figure 9. MCSCF+1 +2 relative energies of the Sc+ + SH2 reaction.

62

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is of 42‘ symmetry, while the o singlet coupled oxygen-metal

electron pair structure

drt

 

is of 4H symmetry.3O We calculate that these two states are
separated by only 7 kcal/mol and that the lower, the 411, has a
calculated bond energy of 57 kcal/mol. This is significantly less than
the triply bonded +ScO(121+) but comparable to the doubly bonded
+Sc0(3A). I ‘

3. .The +Sc-O bond in +ScOH is 108 kcal/mol or 43 kcal/mol
stronger than the bond in +ScO(3A). This is due, primarily, to the

dative interaction with Sc+ of the O 23+2po hybrid induced on 0 when

64

bonded to H. This is in substantial disagreement with the recent5

experimental value of 88 kcal/mol.

4. We calculate three exothermic products of the reaction of Sc+ +
H20. The ion dipole complex Sc+---H20, the oxide - H2 complex

H2"'+SCO, and the insertion product, H-"'Sc-OH. The oxide product is a
consequence of the very strong bond in +ScO(12‘."') which is due, in
large measure, to the O lone pair forming a dative bond to Sc+. The
oxide will not be nearly so exothermic with any other first row
transition element. The insertion product is calculated to be the
global ground state, although by only 3 kcal/mol. The stability of
this product is due, in large measure, to a dative bond between the O

28+2p0’ hybrid on OH and Sc+. As this bond strength is decreased, the

exotherrnicity of the insertion product will decrease.

5. A consequence of the strong “"Sc-OH bond is that the calculated
global minimum in the Sc+ + H20 system is the insertion product H-

+Sc-OH, while the calculated global minimum in the Sc+ + NH3 system
is the electrostatic complex +Sc-"NH3. This situation obtains because

the +ScNH2 bond strength is calculated to be 78 kcal/mol,

substantially smaller than the +ScOH bond strength of 107 kcal/mol.

Since the Sc-H bond energy is similar in both systems (46 kcal/mol
in H-+ScNH2 and 47 kcal/mol in H-+Sc-OH) the insertion product in

H-+Sc-NH2+ lies ~24 kcal/mol above the electrostatic complex while

in H-+ScOH it lies at least 3 kcal/mol below the Sc+mH20 complex.

65

6. The +ScS ground state is a triply bonded species of 12+

symmetry with a De of approximately 97 kcal/mol. The two lowest

triplet excited states are 1t,1t bonded species of 3A and 32+

symmetries with calculated Des of 40 and 28 kcal/mol, respectively.

7. The ground state product for the gas-phase reaction, Sc+ + H28
is expected to be the electrostatic species H2---"'ScS (AB = 34.5

kcal/mol) with the next nearest product, +Sc-"SH2, 23 kcal/mol

higher at 11.4 kcal/mol. The energy of the insertion product, HSc+SH,

is intermediate to H2---+ScS and +ScmSHz but is not a minimum on the

reaction surface.

8. Previous work on +ScOH and +ScNH indicates the slight
difference in size of the ligand has little effect on the Sc-LH bond
strength. Those results suggests that the +Sc-SH and +Sc-PH bond
energies should be comparable. This also suggests that the M-L bond

energies in the pairs

8. 8.”.

(«3 +Ti=_—"SH(32') + (*TiE PH(2A)
6+

(3' (f

°-- . VESH(42') + . V:_-_-_PH(3Z')

k-zs.

6,.

will also be similar.

67

W

1 J.L.Tilson, J.F. Harrison, J. Phys. Chem., 25, 5097 (1991).

2 J.L.Tilson, J.F. Harrison, J. Phys. Chem., 26, 1667 (1992).

3 a) E]. Murad, J. Geophys. Res., 83, 5525 (1978).

b) H. Kang and J.L. Beauchamp, J. Am. Chem. Soc., 1_0_8, 5663
(1986).

c) N. Aristov and PB. Armentrout, J. Am. Chem. Soc., _1_0_6, 4065
(1984).

4 a) E. Murad, J. Chem. Phys, 13, 1381 (1980).

b) H. Kang and J.L. Beauchamp, J. Am. Chem. Soc., 108, 7502
(1986).

c) C.J. Cassady and BS. Freiser, J. Am. Chem. Soc., 1_O_6_, 6176
(1984)

5 T.F. Magnera, D.E. David and J. Michl, J. Am. Chem. Soc., m, 4100
(1989).

6 A. Mavridis, K. Kunze, J.F. Harrison and J. Allison, "Bonding
Energetics In Organometallic Compounds", T.J. Marks Ed., ACS
Symposium Series 428, American Chemical Society, Washington
DC, 1990, Chap. 18.

7 a) M. Rosi and CW. Bauschlicher Jr., J. Chem. Phys., 22, 1876

(1990).
b) M. Rosi and CW. Bauschlicher Jr., J. Chem. Phys., 20, 7264
(1989).

8 A.J.H. Wachters, J. Chem. Phys., 52, 1033 (1970).

9 TH. Dunning Jr., private communication.

10 P.J. Hay, J. Chem. Phys., 6_6_, 4377 (1977).

l 1 RC. Raffenetti, J. Chem. Phys., 5_8_, 4452 (1973).

12 RB. Duijneveldt, "IBM Technical Research Report No. RJ-945",

IBM Research Laboratory, San Jose, CA, 1971.

13

14

15

16

17

18

19

20

21

22

23

24

25

26

68

S. Huzinaga, "Approximate Atomic Functions 11", Research report;
Division of Theoretical Chemistry, Department of Chemistry, The
University of Alberta, 1971.

M. Rivera, J.F. Harrison and A. Alvarado-Swaisgood, J. Phys.
Chem., 94, 6969 (1990).

CE. Moore, Natl. Stand. Ref. Data. Ser., Natl. Bur. Stand. No. 35.

The ARGOS integral program was developed by R.M. Pitzer
(Ohio State University)

The GVB164 program was written by R. Bair (Argonne
National Laboratory).

A description of the UEXP program is given in R. Shepard,
J. Simons and I Shavitt, J. Chem. Phys., 76, 543 (1982).

H. Lischka, R. Shepard, F.B. Brown and I. Shavitt, Int. J. Quant.
Chem. Symp., 15, 91 (1981).

R. Shepard, I. Shavitt, R.M. Pitzer, D.C. Comeau and M. Pepper,
Int. J. Quant. Chem. Symp., a, 149 (1988) and references therein.

.’ —’

J.L. Dunham, Phys. Rev 41 721 (1932).

a) W.A. Goddard III, T.H. Dunning Jr., W.J. Hunt and P. Hay,

J. Acc. Chem. Res., 5, 368 (1973).

b) W.A. Goddard III and LB. Harding, Annu. Rev. Phys. Chem.,
22, 363 (1978).

RS Mulliken, J. Chem. Phys., 25, 1833, 1841, 2338, and 2343
(1955). For a critique see J.O. Noell, Inorg. Chem. 21, 11 (1982).

adapted from "CRC Handbook of Chemistry and Physics”, R.C.
Weast, Ed., CRC Press, Boca Raton, FL, 63rd ed., 1982.

A.E. Alvarado-Swaisgood and J.F. Harrison, J. Phys. Chem., 52,
5198 (1985).

PB. Armentrout, "Bonding Energetics In Organometallic
Compounds", T.J. Marks Ed., ACS Symposium Series 428,
American Chemical Society, Washington DC, 1990, Chap. 2.

27

28

29

30

69

DE. Clemmer, L.S. Sunderlin and PB. Armentrout, J. Phys. Chem.,
%, 3008 (1990).

J.L. Tilson and J.F. Harrison, unpublished results.

J.M. Dyke, B.W.J. Gravenor and A. Morris, J. Phys. Chem., _8_2, 4613
(1985).

J.F. Harrison, J. Phys. Chem., 20, 3313 (1986).

CHAPTER III

CHAPTER III
THE EIECIRONIC AND GEOMETRIC STRUCTURES OF
+ScSe AND +ScSeH

W

The +ScSe and +ScSeH molecules were investigated by
determining the Multiconfiguration self consistent field (MCSCF) and
configuration interaction (MCSCF+1+2) wavefunctions for the 12"‘, 3A
and 323+ states of +ScSe and the 2A and 22* states of +ScSeH. The
ground state +ScSe is a triply bonded species of 12‘." symmetry with a
bond strength of 84 kcal/mol. The 3A and 32+ excited states lie
higher in energy at 31 and 28 kcal/mol, respectively. The +ScSeH
molecule has a 2A ground state nearly degenerate with the excited
22+ state, with both differentially stabilized by formation of the Se-H
bond . This stabilization is consistent with prior work on +ScOH and
+ScSH.

Our focus is on the relative energies of the molecules and the
structure of the valence o orbitals in +ScSeH. Previous theoretical
studies on +ScSH1 and +ScOH2 indicate a differential stabilization
resulting from an induced 0 bond between Sc"‘ and the ligand, L,
formed in concert with the L-H bond. This differential stabilization
results in theoretical predictions of the gas-phase Sc+ + LHZ reaction
that differ from those based on simple bond additivity arguments. In
particular for the reaction Sc+ + 0H2, the ground state was predicted to

be the H+ScOH insertion product, while for the Sc"' + SH2 reaction the

71

72

electrostatic H2---+ScS molecule becomes the ground state. Clearly, the

ground state reaction product is very dependent on the amount of this
stabilization. The results here indicates a definite trend in electronic
structure and in the amount of extra stabilization for these Group VI-
containing molecules.

There are no experimental results for the +ScSe(H) molecules nor

on the reaction Sc+ + SeHz. The only direct experimental data available

for comparison have been the experiments of Magnera, et al.3 on the
Sc+ + 0H2 reaction. We find, however, definite trends in character going

from O to S to Se and are confident in the predictions.
BA I ET

The Scandium and Hydrogen atom basis sets imployed here
have been used before.1

The Selenium basis was the (13s,9p,5d) set from Dunning,4
augmented with a diffuse s (exp = 0.05592), a diffuse p
(exp = 0.0513) and a diffuse d (exp=0.40) function. This set was
contracted to (7s,6p,2d) following Raffenetti.5

W

All ab-initio calculations were done on a Stardent TITAN
computer located in the Michigan State University Chemistry
Department using the Argonne National Laboratory collection of

COLUMBUS6 codes.

73

All density and difference density contours were calculated
with the MSUPLOT codes and all spectroscopic constants were

determined by performing a Dunham7 analysis.

ERAQflNI ENERGIES

Sc+ and H

The Sc+ and H atom energies were calculated before1 and are
collected in Table 1.
Se

The Selenium 3P state was analyzed with MCSCF and MCSCF+1+2

wavefunctions. The MCSCF function was constructed from the in-out
correlation (GVB) of the doubly occupied 4p1tx orbital plus all valence

spin couplings. This results in 5 CSFs of 3B2 symmetry. Inclusion of

all valence single and double substitutions (of 3B2 symmetry) from

the MCSCF reference space results in the 691 CSF MCSCF+1+2. These
energies are collected in Table 1.
SeH

The 211 state of SeH was examined by a 17 configuration 2B1 MCSCF

function. This was constructed from all spin couplings of a GVB(2/4)

function (correlating the 1: bond and the doubly occupied Se 4p1ry
orbital) and a 4,146 CSF MCSCF+1+2 constructed from all valence
single and double substitutions (of 2B1 symmetry) from the MCSCF

reference space. The total energies are listed in Table 1 while the
dissociation energy (De) is collected in Table 2.

74

H28e

The energy and optimized geometry of H28e was computed with a
37 CSF MCSCF function constructed to correlate in a GVB way the two
bonding orbitals and the Se doubly occupied out of plane orbital
followed by all spin couplings. The MCSCF+1+2 function was derived
from all valence single and double substitutions from the MCSCF
reference space and results in 25,979 CSFs. The equilibrium energies

are listed in Table 1 and the dissociation energies in Table 2.

W +&§§

Studies of +ScS1 and +ScO2 indicate that the Sc+ ground 3D state
and excited 3F states (3Fé—3D E 0.59 eV)8 are large contributors to

the wavefunctions. The +ScSe 12* state MCSCF function that includes

contributions from both the Sc+ (3D) and Sc+ (3F) asymptotes can be

constructed from all spin couplings of the GVB(3/6)9 function
‘1' ~ (core)2 (14o2-115o2)(6nx2-v7ux2)(6ny2why?)

and results in 37 configuration state functions (CSFs) under a C2v point
group. The core electrons have been suppressed here for brevity but
are always variationally optimized. The term 1462-21562 represents

the coupling of the Se 4po and Sc+ 4s electrons into a 6 bond and allows

the proper separation at large internuclear distances. Similarly, the

6nxz-v7nx2 and 61:),2-v71ry2 terms represents the 1: bonds with

separation to the Sc 4p1t4 and Sc+ 3dr:l configurations.

75

Table 1: Total Fragment Energies (au)

Fragment Emin (MCSCF) Emin (MCSCF+1+2)
Sc+ 3D(4313d1)a 459.52848 -759.52906
Sc+3132(36n13601)a -759.48576 459.49960
Se 31> 4399.71554 4399.75294
SeH 2r1 4400.31795 440036659
H28e 141 4400.92542 4400.99459
H2 lr:+g(3s/3p)a -1.14813 -l.l6652

H 233 -0.49928

a ref (2)

76

Table 2: Dissociation Energies (kcal/mol)

MCSCF(De) MCSCF+1+2(De) Experimentala
SeH 211 => Sc31> + st 64.3 71.8
H2Se 1A1 :3 SeH21'1 + H23 67.9 80.7
H28e 1A1 => 863? + 2H28 132.6 152.5
H2 l2‘.+g(3s/3p) = 21128 93.8 105.4 103.3

aref(12)

77
The 1t bonds are equivalent at equilibrium but arise from formally

different asymptotic occupations. This MCSCF function allows the two

most important 12+ configurations to mix as illustrated below.

 
 
 
  

. SC(3d:;3d:, or “7145) 8: SO( 49049249: )

0 $134,134) & 8614143494148 )

There are two formally covalent bonds (o,1c) and a it dative bond in
structure 1), while in structure 2) there are two covalent 1r bonds and a
o dative bond possible.

The energy predicted by this function is shown in Figure 1 as a
function of internuclear distance. The analogous +ScO2 and +ScSI

MCSCF energies are shown for comparison. The MCSCF function

predicts a triply bonded species with an equilibrium separation (re) of

4.322 au and a dissociation energy (De) of 69 kcal/mol.

The energies of two low lying +ScSe triplet states were also
computed and are displayed in Figure 1 as a function of internuclear

distance. The +ScSe 3A state was examined with a GVB(2/4) function

constructed from two 1: bonding pairs, one electron in an a] orbital and
the other electron in an az orbital followed by all spin couplings. This
results in 25 CSFs. The predicted re of 4.617 au is 0.3 an longer than in

the 12* state while the De decreases to 16 kcal/mol.

78

50,0 — MCSCF Potential Energy / _

 

30.0” 80* (3829 3d2)+I-(3P9 p4)

 

10.0-

 

.1.
P
cpo

1

AE (kcaI/mol)
23’
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-70.0 -
-90.0 -
-1 10.09 \
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Sc012+
-1 30.0 5 \\ ,I’ "

 

 

 

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
r (au)
Figure 1. MCSCF potential energies of the 12+, 3A and 32+ states of +ScSe. +ScS and +ScO

relative to the ground state asymptote. Energy is in kcal/mol. The atomic structure at the
asymptote is indicated by both atomic symmetry and valence configuration (L-Se, S, and O)

79
The structure has two it bonds, an open shell electron in a Se 4po
orbital and another electron in a Sc+ 3d6_ orbital. The Sc+ 3d6_
occupation forces dissociation to the higher +Sc(3d8_13drt1; 3B2) +
Se(4s24p4; 3P) asymptote as indicated in fig 1.

The +ScSe 32+ state MCSCF function was constructed from a
GVB(2/4) function with two 1: bonding pairs and 2 singly occupied 61]
orbitals. The Sc+ 3d8+ orbitals were eliminated from the calculation to
prevent collapse to the 6+ component of the 3A state. All spin couplings
of this function results in 25 CSFs. The computed De of 11 kcal/mol is
5 kcal/mol less than the 3A state and the re shrinks slightly to 4.542

au. The total energies, Des, res and vibrational frequencies (me) are

collected in Table 3.

MCS§§E+1+2 EUNQIIQN FQR +&:_&

The configuration interaction (MCSCF+1+2) wavefunctions were

constructed by taking all valence single and double substitutions from
the MCSCF reference space. In particular, the 123+ MCSCF+1+2 function

consists of 29,953 CSFs, the 3A of 36,686 CSFs and the 32"“ of 32,962
CSFs. The added correlation lowers the total energies by around 61
mHartrees but has no effect on the state orderings. The. energies as

predicted by the MCSCF+1+2 functions are displayed in Figure 2 as a

function of Sc-Se distance. The computed re and we remain essentially
unchanged while the De's increased to 84 kcal/mol (12+), 31 kcal/mol

(3A) and 28 kcal/mol (32+), respectively. The total energies, Des and

some spectroscopic constants are collected in Table 3.

80

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MCSCF+1+2 Potential Energy -

80+ (332, 3d2)+L(3P, p4)

 

Sc+ (so, 4s‘3d‘)+L(3P, p4)

 

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Flame 2. MCSCF+1 +2 potential energies of the 122+, 3A and 32+ states of +ScSe, +ScS and

+Sc0 relative to the ground state asymptote. Energy is in kcal/mol. The atomic structure at the
asymptote is indicated by both atomic symmetry and valence configuration (L=Se, S, and O)

82
W“&

A Mulliken10 population analysis was performed on the MCSCF

natural orbitals (NOs) of +ScSe and the results are collected in Table
4. The 123+ state is a triply bonded system with two 1: bonds and one
0 bond. The 1! bonds are polarized towards the Scandium and are
composed of, primarily, Sc+ 3dr: and Se 4p1t orbitals. The polarization
results in approximately 0.18 electrons (per 1: bond) being transfered
to Sc+. The 0 bond is a mixture of the Se 4pc and Sc+ 3do+7t4s
orbitals. The Sc+ 4s enters the a bond with a weight of
approximately 15%, nearly equal to that of the 3do orbital (20%).
The 0 bond is polarized towards Se and results in a transfer of 0.19
electrons from Sc+ to Se. The gross atomic charge distribution in the
12"" state becomes Sc+O-83Se+O-17.

The 3A and 32‘." states are both doubly bonded (mu) species.
The 1: bonds are composed of, primarily, Sc+ 3d1t and Se 4p1t functions
and are moderately polarized with the transfer of approximately
0.16e per 1: bond to Se. There are also two non-bonding electrons in
these systems. One electron populates a Se 4pc orbital and the other
an orbital on 80“. In the 3A state, the Sc+ non-bonded electron

populates a pure Sc+ 3d6_ orbital. This is the lowest energy triplet

with a bond strength of 31 kcal/mol (Table 3) and a Sc+ (12

configuration, in-situ. In the 32." state, the Sc+ non-bonded electron
occupies an orbital of 3do+k4s character and is 3 kcal/mol higher
(MCSCF+1+2) than the 3A (28 vs. 31 kcal/mol, Table 1). At

intermediate separations the triplet coupled Sc+ 4s and Se 4po

electrons will interact repulsively.

83

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84
The 1t bonds are weak and cannot overcome this interaction. The

repulsive energy increases with decreasing internuclear separation

until it intersects the attractive Sc++(3do) + Se' curve and finally

settles into a structure with the Sc+ open shell electron occupying an

d] orbital of, primarily, 3do character.
The populations and NO structures suggests the following

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1) The 1!: bonds are all composed of Sc+ 3d1c and Se 4px orbitals.

2) The ground state 12+ 0 bond is composed of a Se 4pc bonding to

a Sc+ orbital of equal amounts of 4s and 3do.

3) The 3A and 32+ states are both 1m: doubly bonded systems and
each have a non-bonded electron on Sc+ occupying a predominantly

3d type orbital.

The MCSCF orbital populations of +ScSl and +Sc02 are collected
in Tables 5 and 6 for comparison and reveal several trends. The Sc+
4s contribution, in the 12"" states, increases with increasing ligand
size and presumably results from the longer bonding distances. The
radial distribution functions (RDF) for the Sc+ s, p and d orbitals (Is
not included) were plotted (Figure 3) and show that the 3d RDF has a
maximum at approximately 1 an and decreases substantially
between 2 and 4 au. The 4s reaches a maximum at approximately 3.0

au and decreases only slightly by 4 au.

85

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0.0a

88
The increase in Sc+ 4s contribution to the 0 bond at the longer bond
distances reflects the favorable increase in overlap with the ligand
po.
Secondly, as the bond length in the 12+ states is increased, the
Sc+ gross atomic charge decreases. In particular, the Sc+ atomic
charge goes from +1.28 in +ScO,2 to +1.14 in +ScS1 and to +0.85 in

+ScSe. The gross atomic charges are collected in Table 7.

The triplet states all have very similar It bonding structures
with the ligand valence p115 orbitals making a lesser contribution at
the longer equilibrium bond distances. A conspicuous feature of
these data are the orbital population trends in the 321+ states. The 1:
bonds are nearly identical to the corresponding 3A states but the Sc+
4s contribution to the spectator electron structure decreases with

increasing bond distance.

In contrast to the 12"' states where at longer bond lengths a
greater 0 bond overlap occurs with inclusion of the Sc"' 45, the triplet
coupling of the two 0 non-bonded electrons requires this overlap to
be a minimum. In +Sc02 this is accomplished by promoting the Sc"'
spectator electron to a 4s+l3do orbital with approximately 40% 45
character. This hybridization drains charge away from the
internuclear axis. In +ScSe the 3do orbital is apparently small
enough to not significantly interfere with Se at the large equilibrium
separation and so does not hybridize as much. The +ScSl structure is
again intermediate. The orbital populations for selected valence
orbitals of +ScL (L=O,S,Se) are displayed in Figure 4. These data are
collected in a way to allow comparisons amongst the different

studied states and illustrate the trends in charge transfer.

89

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i ........... a ................. 4 ...
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0 0.8-
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08: —°— 32* +Sc ado orbital“
E ’— 21:” +Sc ado orbnaIb
O 02

 

 

 

o.o , T 1
O S Se

Figure 4. Orbital populations of +ScL and +Scl.|~l (L=O,S,Se). The connection of data points is only for association and does not imply a
continuous relationship. S data from Rem). 0 data from Ref(2).
a Orbital population data relative to associated 3A state. b Orbital population data relative to associated 2A state.

91

In the studied triplets, the lowest energy state has a Sc+ (d2)
in-situ configuration (3A) suggesting that in the field of the ligand
the required excitation of Sc+ to the d2 configuration in-situ is offset

by a decrease in the electron repulsions. The mixing of the Sc+ 4s
orbital into the Sc+ spectator electron density in the 32+ states has a
marked effect on the energy. We find that while the 3A bond
energies decrease with increasing ligand size (bond length) the Sc+ 4s

contribution to the 32+ states also decreases, causing the 32+(— 3A

separation to decrease. This separation is the least for +ScSe and
results in the +ScSe 32““ state being more bound than the +ScS 32"”
state. Theoretical results indicate that the 32+<— 3A separation
decreases in the order +ScO (19.9 kcal/mol), +ScS (12.5 kcal/mol) and
+ScSe (3.0 kcal/mol).

+&§§H QENERAL QQNSIDERATION§

The +ScSeH molecule may be formed by bonding Hydrogen to
either of the open shell electrons in triplet +ScSe. Recent theoretical
studies2 have indicated that H-+Sc0 is less bound than +ScOH due to
disruption of an induced sigma bond, and that +ScOH prefers a linear

conformation. These results were applied here by examining the

linear +ScSeH isomer. +ScSeH has three formal bonds (two +Sc-Se 1:
bonds and a Se-H 0 bond) and a singly occupied orbital that carries

the molecular symmetry.

The 2A state MCSCF function was constructed by correlating the

three bonds in a GVB way and placing the final valence electron in an

az orbital (8-).

92
~11 ~ (core)2 28-1(1402-ll502)(61I:x2-v71tx2)(61ty2-v71ty2)

All spin couplings of this function results in 76 CSFs under a C2v
point group. The 22‘." state MCSCF function was constructed by
correlating the two it bonds in a GVB way and allowing all
occupations of the three valence o (a 1) orbitals (Complete Active

Space in the valence 0 space) to insure that no limitations are

imposed on the calculation. The Sc+ 3d5+ orbitals were eliminated
from the calculation to prevent collapse to the 8+ component of the

lower energy 2A state. All spin couplings of this function result in

144 CSFs under a C2v point group.
‘1‘ ~ (core)2 (140,156,16O)(61tx2-V71tx2)(61ty2-V71ty2)

The MCSCF+1+2 wavefunction for each state was constructed by

allowing all valence single and double substitutions from the MCSCF
reference space. This results in 144,982 CSFs (2A) and 166,770 CSFs

(22+).

The 2A state is bound by 118.6 kcal/mol relative to the ground
state atoms and is 3.2 kcal/mol lower in energy than the 22"' state.
The optimized geometries are very similar. In particular, for the 2A
state the +Sc-SeH bond length is 4.567 au (MCSCF+1+2) with a Se-H
length of 2.760 au, while in the 223+ state +Sc-SeH bond contracts a
small amount to 4.464 an and the Se-H bond lengthens to 2.774 an.
The total energies and optimized geometries are collected in Table 8.

Table 9 contains several computed bond energies.

93

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The +ScSeH valence orbital populations are collected in Table 2
and the gross atomic charges in Table 7. The 2A and 22"' states are
similar, with two (Sc+ 3dr: - Se 4pn)1t bonds, a Se-H 0 bond and a
non-bonded electron on Sc+. The 1: bonds are polarized towards Se
with approximately 0.21e (per bond, 2A) transferred to Se. The Se-H
a bond is also polarized and transfers 0.22e to Se from H. This H to

Se charge transfer is similar11 to that computed for SeH and SeH2

and is consistent with the results of +ScOH2 and +ScSH.l The gross

atomic charge distribution for the 2A and 22"' states becomes
Sc+1'10Se'0°32H+0'22.

The non-bonded electron in the 2A ground state occupies a
pure +Sc 3d8_ orbital and indicates that, as with the +ScSe triplet
states, 3F +Sc (3d2) is the preferred in-situ configuration. The 22*
state non-bonded electron is primarily in a +Sc 3do, and the 22'"-
2A separation is 3.2 kcal/mol.

In comparison to the MCSCF populations of +ScSH1 and +ScOH2
(Tables 5 and 6, respectively) we find that as the equilibrium +Sc-LH
bond length increases, the amount of +Sc 43 mixed into the non-
bonded electron orbital decreases. This trend is consistent with the
triplet +ScL results. When H is bonded to the ligand atom, L
(L=O,S,Se), the 1t bonds become slightly more polarized with from
0.2e (+SCS) to 0.5e (+ScSe) additional charge transfered to L from +Sc.
The bonding of H to L causes L to become more negatively charged

and causes an increased polarization of the +Sc atom.

96
Selected valence orbital populations of the +ScLH 22+ states are
collected in Figure 4.

The similarities in 1!: bond and non-bonded electron structures
are illustrated in the DDCs (Figures 5,6 and 7). These contours were
generated by computing the MCSCF total density for the +ScLH
molecules at the equilibrium geometry and subtracting from this the
total density of the indicated +ScL triplets superimposed at the +ScLH
geometry. Positive contours are indicated by solid curves and
negative contours by dashes. In all plots, Sc is at the origin. The lack
of 1: contours in the DDCs indicates that the 1: density in the triplet
+ScL and doublet +ScLH states are nearly equivalent. The non-
bonding Sc+ o electron has a do+7t4s shape (22+—3A images) with
relatively more density perpendicular to the bonding axis in +ScOH.
The bonding of H to +ScL introduces more charge density to the 0
space (Zr-+321+ images).

The structure of the Sc+ spectator electron in the 22"" state is
also illustrated in the contours of the valence NO amplitudes (Figure
8). The large Sc+ 4s component and its effect on +ScOH is evident. In
these systems the Sc+ spectator electron density is seemingly next to
the ligand atom core. In +ScOH this core is only 3.5 an from Sc+ and a
large 4s component is introduced to allow perpendicular

displacement. In +ScSeH, the longer bond length allows significantly

more population of the 80+ 3do orbital.

97

+ScSeH( 22*) Total Density at equilibrium

L L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

' z-aiiis '
8.00 8.00
w r l r w I I I
m z-axis moo w z-axis 104”
+SeSenr 22+)-+ScSe( 3A ) Diff. Density +SeSent 22+)-+ScSe( 32*) Diff. Density
m l I l m l I I
4.00 r T . r '3'“ r r . f
-e.oo z-axrs 10.00 m z-axns 10.00

Flgure 5. MCSCF total density contours (T003) and difference density contours (DDCs) for the 22" state of +Se£eH and the
12+ and 32+ states of +Sese. The 0003 are molecular differences where the +Scse states are subtracted from the +ScSel-l

density. The +ScSe states are at the equilibrium +Se£eH (Sc-Se) geometry. Contour levels range from 0.0025e to 1.289 (T 065)
md ~0.04e b 0.04s (DDCs). Each level diliers by a factor of 2. No zero contours are displayed and negative contours are indimted by dashes.

98

+Sesnt 22+) Total Density at equilibrium
m L l l

 

 

 

 

 

 

 

   

 

 

 

 

 

 

'5‘
go ‘ F
>-
4'm I r 1
40° z-axis ‘1‘”
+ 3 . + 3 + .
ScS( A )Total Densny ScS( 2 )Total Densnty
..m l L I m L l L
‘ ' ‘ i’
t'». :1-
m l I l m 1 l l
m z-axis 1100 Am z-axis “0°

+Sesnr 223“) -+ScS( 3 A ) Diff. Density +Sesnt 22+) - +Scs ( 32*) Diff. Density
3.00 l I l m 4 l 1

 

 

   

 

 

 

 

 

 

0
4m z-axis 110° 4“ z-axis 110°
Figure 6. MCSCF total density contours (T 003) and difference density contours (DDCs) for the 22* auto of +SrSH and the

12* and 32* states of +S$. The 0003 are molecular differemes where the +86 states are subtracted from the +ScSH

density. The +S$ states are at the equilibrium +S$H (Sc-S) geometry. Contour levels range from 0.00259 to 1289 (T DCs)
:yld 004e to 0.04e (0003). Each level differs by a factor of 2. No zero contours are displayed and negative contours are indicated
dashes.

99

+ScOH( 223") Total Density at equilibrium

 

A

A_-
"

 

 

 

m I I —T
4.00 z.axis 11.00
+ 3 . + 3 + -
ScO( A )Total Densnty Sc0( 2 )Total Densnty
If!) I l l m l- l ;

 

 

   

 

 

 

 

 

 

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1
w z-axis 110° 41” z-axis “0°

+ScOH( 22:“) -+Sc0( 3A ) Diff. Density +ScOH( 22+) -+Sc0 ( 32*) Diff. Density
1 l l m J 1 l

 

 

   

 

 

 

 

 

 

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l . \
>e >e
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Jm I r . I '8' r fi. T
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Figure 7. MCSCF total density contours (T003) and difference density contours (DDCs) for the 22* state of +ScOH and the
12" and 32" states of +ScO The 0003 are molecular differences where the +500 states are subtracted from the +ScOH

density. The +800 states are at the equilibrium +ScOH (Sc-0) geometry. Contour levels range from 0.0025e to 1 289 (TDCs)
and -0.049 to 0.04s (DDCs). Each level differs by a factor of 2. No zero contours are displayed and negative contours are indicated
by dashes.

100

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101
W

Figures 9-11 illustrate the energy relationships for the +ScSeH,
+ScSH and +ScOH systems. The +ScLH (L=O,S,Se) 2A - 22"” energy
separation decreases in going from O to S to Se, becoming 3.2
kcal/mol for L=Se. This is consistent with the triplet results and
indicates the Scandium spectator electron has little influence on the
bonding.

An induced 6 bond causes a differential stabilization of these
systems and this stabilization decreases for increasing ligand size. For
example, +ScLH (L=O,S,Se) may dissociate to +ScL + H along the

character-conserving paths:
+ScLH(2A) 4 +ScL(3A)+H and +ScLH(22+) 4 +ScL(32:+)+H.

If bond additivities were applicable to these systems one might
expect that, since the open shell electron on Scandium isn't involved
in bonding, these energies would be the same for a given L and
moreover that these energies would be equal to the L-H bond
strength. In fact, we find a consistent enhancement of the +ScL-H
energies relative to L-H, with the greatest amount of stabilization in
+ScOH.2 The MCSCF+1+2 stabilization was found to be 42 kcal/mol
for +Sc0H,2 16.5 kcal/mol for +ScSH1 and 15.6 kcal/mol for +ScSeH.
This stabilization is caused by a o dative bond formed between
+Sc and L. Bonding of H to L introduces charge density into the +Sc-L
0 space (Figures 5, 6 and 7). In +ScOH2 this 6 bond is strong
(z42kcal/mol) and composed of primarily an O 25+).2po hybrid

102

orbital bonding to a +Sc 3do+4po hybrid orbital (Table 4; 2A). The O
25+12po hybrid is the companion orbital to the O-H bond. In +ScSeH
the Se-H bond is mostly Se 4pc + H ls; therefore the charge available
for bonding to +Sc is primarily from the Se 45. This 43 donates
charge into a +Sc 4s+3do+4po orbital and results in an induced a bond
significantly weaker than in +ScOH. The +ScSH results are
intermediate. This is illustrated in Figure 8 where the orbital
amplitude contours corresponding to the spectator electron on
Scandium, the L-H bonding orbital and the highest doubly occupied s
orbital on L for the 223+ states are presented.

The induced o stabilization has large effects on the chemistry of

these systems. In the gas phase reaction of Sc+ + H20 the ground

state reaction product was computed to be the inserted H+SCOH
species exoergic by 40 kcal/mol.2 In the reaction Sc+ + H28, we

calculated an induced 0 bond stabilization of 16.5 kcal/mol for +ScSH

and determined the ground state reaction product should be the

electrostatic species H2"'+SCS, exoergic by 34.5 kcal/mol.1 H2 is the
intact molecule interacting electrostatically with the ground 12"' state

of +ScS. The +ScSe work presented here suggests that due to the

small induced stabilization in +ScSeH, the ground state of the gas
phase reaction Sc+ + SeH2 should also be the H2---+ScSe species. The

reaction energy can be estimated using the computed H2 bond
strength of 105.4 kcal/mol (Table 2), the calculated +ScSe 12+ bond
strength of 84.6 kcal/mol (Table 3), and the SeH2 dissociation energy
of 152.5 kcal/mol (Table 2). These values indicate the H2°°°+ScSe

reaction product is exoergic by approximately 37.5 kcal/mol.

103

 

 

 

 

 

 

 

 

 

 

+Sc + Se+2H
150.0 -—t I
|— 28.2
31.2
F ll*ScSe(3z+) + 2H
"— + 3
152.5 ScSe( A) +211
_ 71.8
100.0
9 84.6
c _. + V
:5 Sc + SeH-I'll 115.4
“ —-—
o +
5 — +ScSe(]21) +211
>1
50
l- —
8
a 118.6

 

500___ 46.9
. 2 V
— +ScSeH( 2*)+n

+ScSeH(2A) + H

 

Sc++ SeI-Iz

 

 

0.0

 

Figure 9. MCSCF+1+2 relative energies (kcal/mol) of selected Sc+ + SeH2 products.

 

Energy (kcal/mol)

104

+Sc+S+2H

 

27.6

 

 

150.0 40 1
_ ] +ScS(32+) + 2H

— 78'0 $ +ScS(3A) +211

 

 

 

96.9

 

100.0 — -

-— +Sc v+ SH + H
120.4

 

 

+ScS(l£+) + 211 v

 

 

56.6
134.5

 

 

2
50.0 +Scsm 2”) + H y

 

2
+ScSH( A) + H 1.“

 

 

 

86+ + SH2
0.0

 

Figure 10. MCSCF+1+2 relative energies (kcal/mol) of selected 86'” + SH2 products.
Data from Ref(1).

 

Energy (kcal/mol)

105

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+Sc + o + 2H

— A

200.0
__ 46.5

+ 3 +

— 97. V SC“ 2 ) + 2H
— 66.4

150.0 l +Sc0(3A) +211

214.7

_ +Sc + OH+ H

100.0 *-—-
— 146.0
— 100.0
—- *swdz“) + 2n
._ 188.4

50.0
— 2
+Sc OH( 2”) + H
_ 2 . 17.2 205.6
_ +Scom A) + H I
Sc++ O
0.0 H2

 

Figure 11. MCSCF+1+2 relative energies (kcal/mol) of selected Sc+ + SH2 products.

Data from Ref(2).

 

106

The interaction energy for the other significant electrostatic species,
+SC°°°SeH2, is expected to be approximately equal to that computed
for +SC”'SH2 (11.4 kcal/mol)1 and is not expected to be the ground
state. This is based on similarities of the (MCSCF) dipole moments11
computed for SH2 (1.27 D, experimental value12 is 0.97 D) and SeH2
(1.09 D) and the similar structures of SH2 and SeH2.11

SL1 MMARY

l) The ground state of +ScSe is the triply bonded 12"’ state

with a bond strength of 84.6 kcal/mol. The lowest triplet state is the
3A with two 1: bonds and a bond energy of 31.2 kcal/mol, while the

32"“ state is also a 1m: doubly bonded species with a bond energy of

28.7 kcal/mol.

2) The +ScSeH ground state is of 2A symmetry with a formation

energy of 118.6 kcal/mol and a +Sc-SeH bond strength of 47
kcal/mol. The 22+ state differs only in the structure of the Sc

spectator electron and has a formation energy of 115.4 kcal/mol.

The formation of the +ScSe-H bond induces a Sc-Se o dative bond,
which stabilizes the species by 15 kcal/mol. This stabilization is due

to the Se 48 orbital interacting with Sc+.

3) The small induced o stabilization energy computed for

+ScSeH (15 kcal/mol) suggests the gas phase reaction Sc+ + SeHz will
yield the product H2--°+ScSe, exoergic by approximately 37.5

kcal/mol.

108

W

10

ll

12

13

J.L.Tilson, J.F. Harrison, J. Phys. Chem., 26, 1667 (1992).
J.L.Tilson, J.F. Harrison, J. Phys. Chem., 25, 5097 (1991).

T.F. Magnera, D.E. David and J. Michl, J. Am. Chem. Soc., 1Q, 4100
(1989).

T.H. Dunning, Jr., J. Chem. Phys., E, 1382 (1977).
RC. Raffenetti, J. Chem. Phys., L8, 4452 (1973).

R. Shepard, I. Shavitt, R.M. Pitzer, D.C. Comeau and M. Pepper,
Int. J. Quant. Chem. Symp., _2__2_, 149 (1988) and references therein.

J.L. Dunham, Phys. Rev., 4_1_, 721 (1932).

CE. Moore, Natl. Stand. Ref. Data. Ser., Natl. Bur. Stand. No. 35.
Averaged over Mj values.

a) W.A. Goddard III, T.H. Dunning Jr., W.J. Hunt and P. Hay,

J. Acc. Chem. Res., 6, 368 (1973).

b) W.A. Goddard III and LB. Harding, Annu. Rev. Phys. Chem.,
22, 363 (1978).

RS. Mulliken, J. Chem. Phys., 23, 1833, 1841, 2338, and 2343
(1955). For a critique see J.O. Noell, Inorg. Chem. 21, 11 (1982).

J.L. Tilson and J.F. Harrison, unpublished results.

adapted from "CRC Handbook of Chemistry and Physics", R.C.
Weast, Ed., CRC Press, Boca Raton, Fl., 63rd ed., 1982.

K.P. Huber and G. ,Herzberg, "Molecular Spectra and
Molecular Structure", Van Nostrand Reinhold, New York, 1979.

109
14 E.J. Murad, J. Geophys. Res., 83, 5525 (1978).

15 a) M. Rosi and CW. Bauschlicher Jr., J. Chem. Phys., _9_2_, 1876
(1990).

b) M. Rosi and CW. Bauschlicher Jr., J. Chem. Phys., 2Q, 7264
(1989).

APPENDIX A

APPENDIX A

ELECTRONIC STRUCTURE THEORY:
TECHNIQUES

INTRD N

This appendix is included to briefly illustrate and discuss those
quantum mechanical techniques used throughout this work. It is not
intended to be a rigorous and complete study, but rather a general
outline with the appropriate references. The systems of interests are
small molecules and techniques useful in their analysis will be
discussed. In particular self consistent field192(SCF) and some higher
order techniques will be discussed.3‘6 All reported calculations of
molecular structure are of the ab-initio type. This means all results
are from first principles without resort to experimental evidence
with the exception of the nuclear charge. The electronic structure of
molecules can be interpreted as the distribution of electrons in a
system whose atoms are physically close enough to experience
significant and, often, non-classical interactions. These interactions
are generally in addition to the "classical" interactions of
electrostatics and magnetostatics, and result specifically from the
small physical sizes and distances. An example of such interactions
would be the formation of bonds.

The fundamental equation used throughout this thesis is the

Schroedinger equation7 (wave mechanics)

112

,HLMEL
‘at

and its equivalent Heisenberg formulation7 (matrix mechanics)
where if is the hamiltonian and ‘1’ is called the wavefunction. ‘1‘
represents all the characteristics of the system. In general, both fl

and ‘1’ have an explicit time dependence.

WI!

The hamiltonian is an operator that contains all the possible
interactions upon the system and in an isotropic and homogenous
space its eigenvalues may be identified with the total energy.8 If the
hamiltonian contains no "explicit" time dependence, e.g., impinging
electric fields, the time dependence may be separated out and for
stationary states7 may be ignored. This is done, for example, by
application of an integrating factor. When the time dependence has
been separated out the Schroedinger equation reduces to

W=E‘P
where E is the total energy of the system represented by the
wavefunction, ‘P, and is the response to the actions contained in the
hamiltonian H. The energy is a constant.

The hamiltonian operator for any atomic or molecular system
may be written as
n n 1 2. k k n 2.

1 1 Zk
1Vi-kgz212k\7k+22,— 22 223-3140

if E R
i=1 pt 11 k=1p>k kP i=1 k=1

1

"M”

113
Here ri denotes the coordinates of the 1th electron, Rk denotes the

kth nuclear coordinate and Zk the kth nuclear charge. The terms9

-Lv.2

1
2 1 and ~2Mka2 represents the kinetic energy of the ith electron

and kth nucleus, respectively. The masses are in atomic units (au)

loLfl Zk

, and — re resent the
rij Rkp’ rik p

with me (the electron mass) =1 au.
electron-electron, nuclear-nuclear and electron-nuclear coulomb
interactions, respectively, and ’V(t) contains all other possible
interactions, e.g., multipole, relativistic, and external fields.

The typical electronic structure calculation begins by
eliminating all presumably irrelevant or negligible terms from the
hamiltonian. Since most molecular calculations are for determination
of quantities relative to the separated atoms or fragments, e.g., bond
energies, many interactions common to both the molecular and
atomic regimes cancel out and can be ignored. This often means
relativistic terms are not included. The remaining hamiltonian
becomes

1 n l A A k n 2. Zk

l
”=‘2_Vi2 ‘13:: _Vk2+22_,+22 ‘22—-
i=1121v1k i=1j>irij k=l p>kRkp i=1 k=1 1rik

The analytical solution of 9L1‘I’=E‘Il would yield the exact electronic
structure of any non-relativistic, gas-phase molecular system.
Unfortunately, this expression is impossible to solve directly. A still
further approximation is the Born-Qppenheimerl,2 (BO)
approximation. Here the nuclear motion is presumed to be infinitely

slow, relative to the electrons, allowing the electron distribution to be

114

optimal for all geometries. This adiabatic“ approximation assumes
the nuclear and electronic motions are, essentially, independent of
one another and that the wavefunction can, therefore, be written as a

product of terms, viz.,
w {r1006 {R11

‘1’ represents the electronic wavefunction and is a function of the set

of electron coordinates, {r}, and the set of geometry parameters,

{R}. <I>({R}) is a nuclear function describing the vibrations and

rotations of the system. This adiabatic approximation is useful but
can fail for even fairly small systems.12 The BO approximation is
further simplified with the W10 approximation where
the kinetic energy of the nuclei is set identically to zero. The

hamiltonian under these conditions becomes.

fls-z—Viz +22;-,+ 22R ~22lr— (1)
l=1 i=1 j>i 1] k=l p>k kp i=1 k=1 ik

and the Schroedinger equation becomes ZIPP=E‘I’ where E is now a

function of the nuclear geometry, i.e., E = E({ R}) or simply E(R). This

hamiltonian is that typically used in theoretical studies of isolated

molecules and results in the notion of a potential curve.

115

 

  
  

 

   

)

asymptotic region (Ea,3 )

.. Dissociation energy

Total Energy

 

 

equilibrium basin (Emin)
1

 

I ' "

Geometry

In the above example, the total energy, E(R), is plotted relative to a

nuclear coordinate. The lowest energy (Emin) is relative to the given

coordinate while the asymptotic region corresponds to the separated
atom limit (Em). The electronic dissociation energy (Dc) is defined as

De=Eoo+Emin
The hamiltonian, 9f, contains one and two body interactions.

The one body interactions are simply the hamiltonians for a one-

electron system, viz
n l n 7. Zk
- )3 5V? - ); }-_'
i=1 i=1 k=l 1k

These hamiltonians have analytical solutions which are known.10
I]

n
The electron-electron terms, 2 2 'r—j , are two body interactions

i=1 j>i 1]
describing the inter-electron (Coulomb) forces. The two body
. . 7‘ 1 Z Zk .
nuclear-nuclear 1nteractlons, 2 2 R , are by constructlon a
_ k
k-l p>k P

parameter of the electronic wavefunction and so may be excluded

from the hamiltonian and simply specified for a given molecular

116

geometry. The electronic hamiltonian, He, is written as the sum .‘HC 5
fi(l) + g( 1,2) where li(l) contains the hydrogenic (one body) terms
and g(1,2) the (two body) electron-electron repulsion terms. For a
hamiltonian in this form the total electronic (BO) energy may be

written as

x 1 k
E(R)=Ee+2 )2 R (2)
k=1p>k kP

where Ec is the solution of flc‘PzEe‘P and the nuclear-nuclear terms
are added to the total energy ad-hoc. Unless otherwise specified, the
nuclear-nuclear terms will be assumed folded into the total energy

for the remainder of this discussion.
N WA N

The electronic hamiltonian, He, contains all the information of
the molecular system under study. If the electronic Schrodinger
equation, flc‘P=Ec‘P, could be solved (or equivalently the hamiltonian
matrix diagonalized) the wavefunction ‘1’ would tell us the exact
electronic distribution and energy, the excited states, and allow us to
determine the equilibrium geometry. Unfortunately this cannot be
directly done. The inability to solve the Schrodinger equation is
caused by the two-body electron-electron terms in the hamiltonian.

If, for example, 30,2) were zero and electron spin were not

considered, 9-[6‘11=Ec‘1’ would become

117

n l n k Zk
(-2 -V-2 - 2 2 .— )4:an
i=1 2 1 i=1 k=lr1k

and the exact solution (‘1’) would be a simple product of hydrogenic

wavefunctions of the form
‘P=(¢1(1)¢2(2).--¢n(n)).

This is a rigorous result with 0 an atomic or molecular orbital (MO).
In this example, the spin-less electrons move independently of one
another. Obviously, this approximation does not describe the electron

correlation effects known to be important to most molecules. The

addition of the g(1,2) term allows for electron motion that is

correlated, but also precludes an analytic solution. Since the
hamiltonian of interest cannot be solved, the solution must be

approximated in some way.

The energy of the Schrodinger equation, flc‘P=Ec‘P, can be

expressed as

E_<‘PI%I‘P>
‘ <‘P1‘I’> '

Where the bra-ket notation stands for

<WIyaIW>=jmedc

118

with integration assumed over all variables (1:) and over all space. ‘I‘*
means the complex conjugate of ‘P. The set of solution wavefunctions
(‘Pin all of its possible states) are orthogonal and can be interpreted
as vectors that span the space of the hamiltonian.7 As in a three
dimensional coordinate system, it is desirable to have these

independent and orthogonal vectors normalized. The normalization

criterion for ‘1' can be written as <‘PI‘I' >=1. For a normalized
wavefunction, the energy expression becomes =<‘Pl:1-[cl‘I’> and is the
integral form of the Schrodinger eigenvalue equation.

The hamiltonian operator is a Hermitian operator.7 This
property causes all eigenvalues to be real and equally important,
that a variation principle1 applies. In particular, the variation
principle states that for a function <1> used as an approximation to the
exact function ‘I’ in the equation EexaCt=<‘P|9{cl‘P>, the energy E” 2
Ben“. This also means that if E”: Eexact then <I>=‘I’. This suggests that
if an arbitrary trial function (with the proper asymptotic and
topological behavior) were selected and the energy (Etrial) computed
we would obtain Etrial 2 EexaCt. This affords a technique for
determining the true wavefunction of a species. In essence we can
select a function, calculate the energy, select a new function and
calculate its energy and compare the two. The function yielding the
lower energy is closer to the true function.

The intrinsic spin of electrons is an experimental fact and must
be incorporated into any wavefunction containing electrons.
Moreover, statistical theories13 of identical particles indicate that
wavefunctions comprised of bosons (zero or integer spin particles)

must be symmetric upon the interchange of particle coordinates

119

(including spin coordinates) while a wavefunction comprised of
fermions (e.g., electrons) must be anti-symmetric upon coordinate
interchange. This intrinsic requirement for electrons is called the
Pauli exelttsien prineiplel’lo and has profound effects on the
electronic structure of molecules. In essence, this theory prevents
two identical electrons from occupying the same region of space at
the same time and is responsible for electron correlation in addition
to the usual Coulomb repulsions. This requirement must also be
incorporated into any electronic wavefunction. Noteworthy is that a
spin based mechanics is not unique. A completely spin free quantum
mechanical approach has been developed by Matsen14 using
permutation operators.

The hamiltonian, fife, contains two terms; 5(1) and 30,2). The
necessary spin conditions, not part of this hamiltonian, must be
incorporated into the wavefunction ad-hoc and are justified only in
the usefulness of the results. The inclusion of spin into the
wavefunction is often done with spia-erbitals.2,10 We assume the
motion of each electron is represented by a spatial function, 0, with a
spin of 1/2 an and a spin projection in either the up (a) or down (8)
directions. The complete function for the ith electron is then written
as ¢(i)ot or ¢(i)B with <a|a>= <BIB>=1 and <alB>=O. ¢(i) will be
considered a spin function (orbital) unless otherwise specified. The
final wavefunction must now satisfy the statistics of fermions, viz,
the Pauli principle. If the wavefunction were simply a product of
spatial functions this property can be introduced by means of a

Slate; gletetgminant.lo Written in this way, the function is always

antisymmetric upon coordinate interchange. In particular, for an n-

120
electron normalized wavefunction comprised of the spin orbitals, 0,

the Slater determinant becomes

"41(1) 41(2) . . 4101) "
1 ¢2(1) 112(2) - - ¢2(n)

 

 

-¢n(l) ¢n(2) - - (1,1101) .1

where the brackets indicate a determinant expansion. This
wavefunction is often more succinctly expressed as ‘P
=fl(¢1(l),¢2(2)...¢n(n)), where only the diagonal terms of the
determinant are displayed and fl is called the antisymmetrizing
operator. Finally, the energy for the molecular system using a Slater

determinant wavefunction becomes

 

 

 

 

_¢1(1)¢1(2)..¢1(n)— '¢l(l)¢1(2)..¢1(n)‘
1 42(1) 42(2) . .4201) 1 02(1)¢2(2) . .4201)
E=<§. . lflelfi. . >
_¢n(1>¢n<2) . 4,4101 _¢n(1)¢n(2) . . 4.01) .
with the total energy computed as
n n
E=.2<hi>+ 2 (2111' Kg). (3)

[=1 i=1>j
< hi > is the one electron energy. Jij is the eealembie energy between
electrons i and j and Kij is the non-classical exehange energy between

electrons i and j. This can be equivalently written as

121

n n
E=2<ili>+ 2(2<ij|ij>-<ij|ji>) (4)

1=l i=1>j

where < ij I ij > is the coulomb energy and < ij l ji > the exchange
energy. This notation implies a specific electron order. In particular,

for the coulomb energy

 

1'12

.. .. Jdv(1)dv(2) ¢i(1)¢i(2) ¢i(1)¢i(2)
<1jl1j>= - -

while for the exchange energy

 

.. .. Idv<l>dv(2) 41(1)4i(2) ¢j(1)¢i(2)
< 1j |j1> =

1'12

Using the hamiltonian, the variation and the pauli-exclusion

principles, the approximate wavefunction for a molecular system

may now be obtained. First, if the two-body hamiltonian term, 50,2),
were small relative to the H0) term, the n-electron wavefunction
would be almost an antisymmetrized product of spin orbitals (MOs),

i.e.,

"41(1) 41(2) . . 41(n) '
1 42(1)42(2) . .4201)

 

 

_4n(1) 4n(2) I 3 4,01) ..

122
or simply 2(41(1).42(2)...4n(n)).
The form of these orbitals is not yet specified. The total energy is
now a function of the unknown orbitals E(ol, ¢2,,,, ¢n)- The variation
principle states that the lowest energy would be obtained with the

exact function. This also implies that with the orbital wavefunction

model (Independent Partiele Medel or IPM) the lowest energy will

be obtained when the best orbitals are selected for ‘I’. There are
many ways to select these orbitals, but the best way to begin is with
the Hartree-Fock142 (HF) method.

The HF method begins with the expression

Etrial = < fi(¢1(1).¢2(2)---¢n(n)) l He l .fl(¢1(1),¢2(2)...¢n(n)) >-

We then perform a functional derivative of E with respect to the

orbitals and set them equal to zero.
8E

3(1)“

This expression will be satisfied when the orbitals, on, are the

 

=0

optimum orbitals. The constraints on this equation are that <¢i|¢j>=5ij

and that variations in the orbitals themselves are orthogonal to all

orbitals (<8¢i|¢j>=0). This procedure results in the HF equation

7¢i=ei¢i

where f]: is the Fock operator, ¢i is the ith optimized orbital and 8 is

the ith Lagrangian multiplier9 often associated with the orbital (one-

electron) energy. Solution of this equation yields the best possible

123

orbitals within the given IPM ansatz. The Fock operator for a closed

shell system of n electrons has the form:

2

2
is fi(1)+22]j(1)-K,-(1).
j=l

fi(1) is called the one-electron operator and includes the kinetic
energy of the electron and the interactions of this electron with all
nuclei.

74 Zk

1
fi(1)= --V2 - >2 —
2 k=1rk

]j(l) is called the coulomb operator and has the form

 

dV(2)¢‘(2)¢'(2)
J1,“ _J 1 .I

— r12

79-(1) is the exchange operator and has the form

 

d 2 '2 T -2
79(1)=JV()¢J() 124.()

1'12

where {P12 is a permutation operator.

The orbitals, (4,, are spatial orbitals. The explicit representation

of the electron spin has been integrated out yielding the ]j and K]-

124

operators. The one electron energy is obtained from the HF equation

as
<¢jlff|¢i>=8i5ij

Both the ,7 and 7C operators depend upon the solution orbitals ¢j and

therefore the equation must be solved self consistently.
E A N

The HF equations are coupled integro-differential equations. A
numerical solution of these equations would yield the exact solution
called the Hartree-Fock solution or the HF limit. The non-linear
character of these equations makes a numerical solution very
difficult to obtain. Moreover, the results could not be used to
interpret the Chemistry of the system. A method amenable to larger
molecules is the Self Censistent Field142 method (SCF). General
mathematical theorems state that a function may be expanded into a
complete set of basis functions so long as the global properties are
the same.9 For example, expansion into polynomials and exponentials
are frequently used in chemical applications. More true than not,
most equations used in the physical sciences tend to result from the
expansion of an unknown function into something else, e.g., Hooke's
law as a truncated power series expansion.

This technique is applied to the SCF approach by expanding the
unknown orbitals, 4), into a set of basis fenetiens (basis set). This

discretizes the HF problem by specifying the form of the orbitals to

125
within some unknown parameters. These expansien eeeffieients are

then to be determined self consistently. In particular we can expand

the spatial orbital ¢i into m basis functions as
m
¢i = Eciuxit
u=l

Where X11 is the nth basis function and Ci”, the expansion coefficient.

Substituting the orbital expansions into the HF equation and writing

it in a matrix form yields

FCi=8iSCi

Where Ci is an m component column vector containing the expansion
Cil
Ci2

coefficients, . 8 is the one-electron energy, S is the

Cim

(X11 X12 "le \
(non diagonal) basis function overlap matrix, x21 X22 ”XZm
\Xml Xm2 - - xmm )

and, F, is the Fock matrix with the elements < Xm l f I xm' >.

 

 

The solution of the HF equation yields the optimized orbitals in terms
of C1. These orbitals may then be used to compute the total energy.
Specifically, the complete set into which the orbitals are
expanded must be of infinite size. This is not possible in practice and
so a finite size basis is selected. The truncation of the basis set causes

the energy to increase relative to the HF solution. This deviation can

126

be minimized by judicious choice of type and size of the basis set.
Many basis functions have been examined but the Guassian Type
Functions (GTF) have found the greatest use.15 The use of GTF‘s and
the algebraic solution to the electron integrals (< ij I ij >) was first
outlined by Boys.16 In the GTF expansion, functions of the form Ci Ni
r118 EXP(otir2) are used to approximate the orbitals. Parameters in
these basis functions are the exponents (ai) which are selected and
the expansion coefficients (Ci) which are to be determined in the SCF
calculation. Ni is a normalization constant and rng is used to ensure
the proper radial behavior. The basis functions only depend upon
the radial coordinate (r). The angular coordinates are used to
partition the Fock matrix into symmetry blocks749 and to select
which basis functions can be combined together. They do not
explicitly enter the numerical stage of the calculation. GTF functions
have been tabulated for most of the atoms, and procedures for
extending these basis sets to the molecular environment have been

outlined.1 5

W

The SCF procedure is a useful and often applied technique in
the determination of molecular structure. It is especially useful for
systems with a closed shell structure (all orbitals doubly occupied)
and that contain no transition metals. SCF theories cannot model the
dissociation of bonds nor other structural correlations important to
most systems. Moreover, for transition metal systems with low lying

excited states, the SCF can not include the important near degeneracy

127

effects. There are several methods for extending beyond the SCF
method. These variational techniques are based upon the same
general idea.

The SCF calculation starts with a single-determinant
wavefunction occupied with n electrons. The orbitals are expanded
into m basis functions with usually n < m. Since we expand the
orbitals into m basis functions and only (at most) n are occupied,
there are m-n unoccupied "orbitals" left over. These are called virtual
Male. Extensions to the SCF procedure use these orbitals to include
correlation in molecular problem. Instead of using a single SCF
determinant wavefunction, we start with a wavefunction consisting
of many determinants. In these calculations all functions must carry
the correct spin and angular momentum, and while a single
determinant may not have the correct symmetry, several taken
together might. A collection of determinants in this way is called a
eenfigeratien state funetien (CSF). Computationally, however, the
programs can often optimize the wavefunction using either CSFs or
determinants. The additional determinants (configurations) are
constructed by "exciting" one or more electrons in the SCF
wavefunction to virtual orbitals. For example in a system of n
electrons populating n orbitals and m basis functions with (m-n)
virtual orbitals we can create an "excited" determinant in the

following way:

‘I’SCF = fl(¢1(1),¢2(2)...¢n(n)) 9 ‘1” = fl(¢1(1),¢n+1(2)---¢n(n))

128

Where now electron 2 occupies the (n+1)th orbital (virtual). This is
an example of a single exeitation. Multiple simultaneous excitations
may also be included. The maximum excitation level is dictated by
the number of electrons and the size of the basis set. A typical

wavefunction with single, double, etc. excitations is shown below.
‘1' = WSCF + 2 Ci Wi + E Cij Wij + .szin Wijk +
1 1] l]

The wavefunction, ‘P, is an expansion in terms of electron excitation

level with (typically) the SCF determinant as the first and usually
dominant term. The initial function from which excitations are taken
is often called the referenee speee. The additional determinants
reflect the level of excitation: l, 2, or 3 electrons, etc. These
excitations can be as few as 2 up through the number of electrons in
the system.

The selection of these additional determinants and choice in
orbitals (MOS), called the n 'l i, is what generally
distinguishes the different available ab-initio techniques.
Configuration Interaction324 (Cl), Generalized Valence Bond7 (GVB),
Coupled Cluster17 (CC), etc. techniques are theoretically different
procedures for adding determinants to the wavefunction. Many of
these techniques require previously determined MOs from which
these determinants are constructed. The variation principle is then
used to optimize the expansion coefficients.

The Multiconfigurational SCF5 (MCSCF) technique is a

procedure for optimizing the expansion coefficients and the orbitals

129

themselves. In practice the MCSCF requires fewer determinants
relative to the CI for a given accuracy. In the limit of an infinite basis
set and full excitations (all electrons) the MCSCF = CI. The inclusion
of correlation is not limited to adding determinants. The success of
the GVB technique is that while multiple determinants are included,
the orbital basis is not initially orthogonal but can be transformed to
an orthogonal set. This allows the GVB to carry fewer determinants
while still properly describing important correlations in a molecule.
Some workers use an orbital basis that is non-orthogonal.18

The studies performed in this thesis typically began with a
MCSCF calculation. This calculation is relatively compact with only a
few determinants selected and allows an interpretation in terms of
orbitals. The energetics were computed with a MCSCF+1+2 function.
In this configuration interaction calculation, the first term in the
wavefunction expansion is not the SCF determinant but the many-
determinant MCSCF (MCSCF reference space). All single and double
excitations into excited MCSCF configurations were then selected.
Finally, the GVB+1+2 calculations were constructed from all single

and double excitations from a GVB reference space.

WES

131
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133

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APPENDIX B

 

APPENDIX B

LISTING OF PUBLICATIONS

Included here is a listing of publications resulting from this

dissertation.

l. "The Electronic and Geometric Structures of Products of the
Sc+ + H28 Reaction" J. L. Tilson and J. F. Harrison, J. Phys. Chem.,

26,1667, (1992).

2. "The Electronic and Geometric Structures of Products of the
Sc+ + H20 Reaction" J. L. Tilson and J. F. Harrison, J. Phys. Chem.,

25,5097, (1991 ).

3. "The Electronic and Geometric Structures of +ScSe and +ScSeH"
J. L. Tilson and J. F. Harrison, accepted J. Phys. Chem.

135