Ab Initio Relativistic Quantum Chemistry of Third ... - ACS Publications

Chapter 21

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Ab Initio Relativistic Quantum Chemistry of Third-Row Transition Elements and Actinides G. L. Malli Department of Chemistry, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada Ab initio fully relativistic Dirac-Fock (DF) self-consistent field (SCF) calculations for a large number of diatomics of sixth-row elements and actinides (Th, U, Pu) are reported. Our methodology has been delineated recently in the first ab initio Dirac-Fock (DF) as well as relativistic configuration interaction (RCI) calculations (with DF as reference) for AuH. The effects of both relativity and electron correlation on bonding, dissociation energy, bond length, vibrational frequency, dipole moment, etc., of diatomic species involving atoms with Z≥75 are discussed. It is concluded that the relativistic (via the Dirac equation) and electron correlation effects must be included in a l l reliable ab initio calculations for these systems. Less rigorous and approximate methods are unreliable even though they can yield results which fortuitously agree with the experimental results. The

theory

basic

of

relativity

foundations

of

and quantum m e c h a n i c s

theoretical

physics.

that

quantum m e c h a n i c s b a s e d upon the

used

for

decades

physicists special Since

assumes

i.e.,

theory the

relativity different

from

Schrôdinger's

a

electrons

Indeed,

nuclear

system with a priori

in

light

Z>70.

suggest

the

of

to

by be

of

bodies

light)

as

two

known has

structure

it

been by

is

non-

does not

obey

assumes),

it

is

safe

to

(NRQM)

the

simplest

These for

simple

velocity

Bohr model

that

this

will

(NR)

(which

mechanics

conclude

to be

study

of

to

of

if

that

one-electron for

(which the

use

behaviour

appropriate

happen

understanding

that

the

comparable

considerations would

a proper

special

significantly

Newtonian

and m o l e c u l e s would not

Ζ predicts

a c c o r d i n g to

is

non-relativistic infinite')

systems moved a t

that

the

well

Schrôdinger equation

fast-moving

velocity

quantum m e c h a n i c s

even

charge

and m o l e c u l a r

Lorentz-invariant

predicted

atoms

these

not

also

Schrôdinger equation

atomic

However,

constitute is

relativity.

equation

of

light.

is

finite

of

non-relativistic

with

of

that

of

in

it

behaviour

(with

velocity

electrons

investigate

and c h e m i s t s .

relativistic; the

to

It

an

the of

atom atomic

therefore the

electronic

0097-6156/89A)394-0291$06.00/0 ο 1989 American Chemical Society

Salahub and Zerner; The Challenge of d and f Electrons ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

292 structure (Z>75)

of

the

mechanics

atoms

which

relativity

is

is

in

involving

quantum m e c h a n i c s

conformity

mandatory.

with

Various

and

special

theory

Dirac

who

in

d i s c o v e r e d the

(4)

1928

(linear

in

only

Lorentz-invariant

of

is

momentum

electron

predictions Lamb and

shift forms

atomic that for

agree

the

(and H

basis

(6)

+

be

cannot

made

remain

theory

with

the

is

being

for of

the

for

decades because

not

for

to

(in

the

electrons

of

first

in

calculation

on the

relativity

and

electrons

of

shown b y

compared to

it

known

(7)

that

a t o m was

out

due

to

of

that an

came

the

the

electrons, total

angular

the

notation

i

(7)

and i

the

electrons

case

behaviour

who

of

i-1/2

Swirles

no

valence

that

of

valence

that

whereas

increased were

and

5d)

effect,

the (8).

-

which

d and

relativistic

and j

i+1/2,

A few

Hg

years

weakening

orbitals

s electrons, for

the

more

(i.e., for

Mayers

in

contracts thereby

important

Since

atomic

charge

indirect

mean

electron,

5d e l e c t r o n s

these

strongly

values.

5d e l e c t r o n s

relativity

the

the

less

larger

the

the

by

one-electron

the

unimportant

designate

introduced by

all

nuclear

This

-

was

of

relativistic

found

was

energy

viz.

b e i n g most

momentum j

dynamics

there

H E A V Y ATOMS

occupied by

in

the

primarily

their

the

orbitals

binding

shielding of

and

chemists

decreased s i g n i f i c a n t l y ;

effect,

although

s75) of

where

light,

relativity. with

their

The

these

two

smaller

in

third-row

transition

ignored orbit

in

the

atoms

of

atoms

of

accurate

for

approximate out

there ab

capable

of

for

basis

function which

symmetry far

of

discuss

a

(15). easily

of

DFAOs

This

insight

the

is

the

the

be

the

spin-

the

«

number

of but

significant

computational

challenge

involving

for

of

and

5d,

this

the

electron

6d and

5f

symposium.

orbital.

the

basis

set

atom have has

of

advantage

of

avoiding certain

admixture

into

the

equation,

the

the

in

used by M a l l i any

technical

and

type

of

the

all

calculations

the

Dirac-Fock

that

of

the

has

the

the

the

negative

describe

the

behaviour

the

it

basis

program

maximizing

the

bonding can

be

and

additional

difficulties

of

full

atomic

b e e n u s e d as

concepts t r a d i t i o n a l l y It

Program

initio

and has

advantage

quantum c h e m i s t s .

wavefunction

ab

Oxford Dirac-Fock

wavefunction

terms

In

numerical

of

carried

such

can handle

on a n u c l e u s

atomic RIP,

for

Integrals

recently

The RIP

been

calculations

methods

relativistic

AuH.

isolated

which

benchmark

called Relativistic

involving

energy of

the

solutions

of

positrons

(16-17). In

computations

with

relativistic

molecular

DFAOs

constituent

is

very

These and

of

the close

orbitals

the

to

unity

and the

electrons

not

RIP,

orbitals atoms

while

(RMOs) in

the

electrons included in

that

is

found that

they the

many

are

almost

the

coefficient

remaining core

identical of

coefficients

contain are

are

i

in

relativistic

relativity of

j

significant

orbitals)

the

double

d i a t o m i c s was

used by

Dirac

as

the

the

longer

atom m o l e c u l e s have

fully

successfully

the

that of

become v e r y

theme

computed u s i n g the

in

so

very

only

relativistic

centered

molecular

interpreted

main

heavy

fully

atom

each

are

are

Therefore

elements

the

of

sub-levels with

outer

of

result

addition,

in

effects

effects

bonding in

choice of

into

for

relativistic

(DFAO)

The

be

program,

performed with

orbitals set.

to

not

a are

orbitals

Furthermore,

formidable

the

two

electrons

large,

as

c a n no

In

and hence

an obvious need

heavy

(14)

2

elements.

performing

Pyper

actinides

importance

H E A V Y ATOM M O L E C U L E S A l t h o u g h a v a r i e t y

initio

A computer

calculations

Z

in

to by

becoming

stabilized

atomic

into

that

nucleus

compared

hence,

are

fact

significantly

electrons

elements.

as

heavy

to

the

heavy

decreases

f

outer

(Z>75).

both

IN

is

d and

orbitals

substantially

an e l e c t r o n

s>p>d>d.

level

the

systems of

EFFECTS

reliable

systems.

so

face

the

very

effect

and the

very

to

appreciable

correlation

calculations

(10-13),

(RIP),

is

electron

which happens

RELATIVISTIC

using

elements

due

very

and 5f

an i>0

are

is

these

i n v o l v i n g heavy

correlation

effect

affected

the

increase

calculation of

electrons

of

valence

near

order

6d)

destabilization

time

momentum o f

elements

c h e m i s t s must

the

effect.

electrons)

2

s and ρ e l e c t r o n s

(and

elements

also

of

especially valence

the

systems

quantum

the

splits

heavy

heavy

addition

amount

effects,

chemistry

(and

/

relativistic

in

These e f f e c t s

electrons

for

5d

interaction

±1/2.

in

the

1

relativistic

the

electrons

atom

indirect

and p

angular

relativistic

destabilized while

(s

therefore

direct

increasing total

the

velocity

and are

progressively


direct

uranium

to

occupying penetrating

momenta

spend an a p p r e c i a b l e

that

for

the

due

electrons

angular

s t a b i l i z e d by they

293

Chemistry of Third-Row Transition Elements and Actinides

termed

to

one

are

the DFAO

small.

the

core,

accommodated i n


the


294 valence

RMOs c o n s t i t u t i n g

general, The b a s i s four

thevalence

a r e composed o f s e v e r a l s e tused t o express

different (I)

types

DFAOs

wavefunction

DFAOs

which,

i n

o f theconstituent

thevalence

atoms.

RMOs c a n c o n s i s t o f u p t o

o f functions (14):

that

aren o tcompletely

filled

i n the isolated

atoms ; (II)

DFAOs be

completely

filled

significantly

i n the isolated

affected

atoms

by the formation

that

might

o f the

molecule ; (III)

E x c i t e d DFAOs u n o c c u p i e d i n t h e i s o l a t e d


might

contribute

significantly

atoms

b u t which

to the formation

of the

molecule ; (IV)

Functions, needed

called

augmenting

to describe

valence

charge

small

functions

residual

distribution

(AF),

which a r e

distortions

onformation

of the

o f the

molecule. Although (I), the

s e t must

needed

same a t o m , basis,

A F s make features

called

selected

functions

atom

lost

o f type

thecoefficients

predicting

have

incorrect RCI

energy, The

techniques

theresults

well

importance

of

calculations significant

This

by using the

i s achieved by using the to construct

wavefunctions

properties

other

feature

as the computational

(14) and because

However,

o f space

we p r e s e n t

relativistic

a summary

D i r a c - F o c k SCF a s basis set

f o r AuH t o d e m o n s t r a t e t h e

as well

as electron

transition

correlation

elements. RELATIVISTIC

CALCULATIONS

binding

t h e 6s-6p

FOR_AuH

hybridization

o f thebonding so that set.

energy

and

distributions, etc.

t h e I s DFAO o f t h e h y d r o g e n a t o m .

( 1 4 ) show t h a t

more

can

than the

s e t f o r AuH c o n s i s t s o f t h e 5 d , 5 d a n d

the chemical basis

experimental

ionicities and

a r e remedied

D F S C F E B S AND

(RCI)

energy

INTERACTION

o u tu s i n g a n extended

functions

FULLY RELATIVISTIC

t h e g o l d atom p l u s

enter

carried

to the

o f thebinding

molecular

as well

fully

o f third-row

CONFIGURATION INTERACTION chemical basis

contribute

charge

here.

ab i n i t i o

of therelativistic

i n diatomics

INITIO

moments,

(III), f o r

i s deduced b y

a n RMO c a l c u l a t i o n

methodology

basis

importance o f

CONFIGURATION

(14).

molecular

i n bonding

i n the gold

set.

described elsewhere

o f our

The

t h e RMO a n d R C I

be repeated

o f 27 v a l e n c e

effects

The

from Both

as RCIc a l c u l a t i o n s

(EBS)

AB

cannot

they

o f overestimating

dipole

o f our

arefully

limitations

to the Hence t h e

played

5 d DFAOs

t o type

thefraction

calculations

e.g.,

details

(14).

d i s s o c i a t i o n products

wavefunctions.

role

i n t h e g o l d atom,

thebasis

been used to calculate

total

t o t h e RMOs.

_The

belonging

THE R E L A T I V I S T I C

u n o c c u p i e d RMOs r e s u l t i n g accurate

Since

t o DFAOs b e l o n g i n g

t h e 5d and

with which

from

o f RMO t h e o r y

t o perform

o f type

work.

consisting o f functions

(III).

elsewhere

orbitals

6p DFAOs

them

(CB),

and

o f which

i n detail

involving

by excluding

accurate

contributions

RMOs a n d b y c a l c u l a t i n g

defects

RIP

(II)

(II),

CORRELATION E F F E C T S : The

small

(I),

t h e 6p a n d

examining valence

only

types

arediscussed

example,

a l l the functions

o f thebonding can be understood b y u s i n g a

the chemical basis

from

hybridization

of

include

f o rquantitatively

AFs a r e constructed t o be orthogonal

essential

by

thebasis

AFs a r e only

However,

6 s DFAOs Our

6p a n d 6p DFAOs

since

DFSCF

i s n o ta do n o t

1.0 eV o u t _ o f t h e

o f 3 . 3 6 eV a r e l o s t

i f t h e 5 d and 5d


21.

MALLI

DFAOs two in


and t h e i r

DFAOs which

energy

value.

The

5 d DFAOs

in

ten of

electrons 0.663

of

The

interaction

were

relativistic (the bond

shows

(R )

(a> ).

2102

for

R

the

non-relativistic

are

in

2.879

serious au and

1

improved wavefunctions 2.963

au and Our

2102

results

molecular

cm"

of

even

are

the

au and

the

R

features

chemical basis

of

1745

;

the

is

most

readily

from

the

a RIP

computation.

three

hybrid

orbital

in

interaction is,

which

is

with

however,

with α

the

responsible the

found

erroneously

interaction 5d-6s

whilst energy

the

direct

gap

of

polarity

AuH.

for

two

from p r e v i o u s

an approximate play

conclusion

is

incorrect

be

left

in

SCF c a l c u l a t i o n our

relativistic

the

5d-6s

a the

hybridization

hybridization calculation

solely

Is

greater

in

configuration

the

core

of

potential

relativity

role of

AuH. of

in

the

our Our

6s

to

interaction

(ECP)

that

eV

for

5d

5d 1 0

basis the

The

5d-6s in

the

greater

results Hay

et

al.

calculation

the

that

extended

1.682

the

e.g.,

chemical bond

result

DFAO. the

predicts

contrary

the

DFAOs

hybridization

calculations;

of

greater

because 5d

non-

degree

The

arises

the

The

the

significantly

5d-6s

from

DFAOs.

5d and

NR w a v e f u n c t i o n

rigorous

a value

the

stabilizes

reduces

the

i n view

core

formed

orbital 5d-6s

its

nonbonding o r b i t a l s .

effect

the

dominant

predicts

a π gold

important

hydrogen

destabilizes

treatment

to

to a

from

which

non-relativistic

and

an e f f e c t i v e

appear

the

conclusions are

less

of

bonding through

bond i s

effects to

two

reducing

This

revealed

understood

demonstrated

show t h a t

the

the

Moreover,

concluded from

cannot

of

the

the

relativity.

be

seriously underestimates

of

effect

above-mentioned

reported using

one

leading

case.

Our

also

relativistic

these

thus

6s

third

limit,

DFAO. in

that

gold

importance

relativistic

not

in

relativistic

combination

(11)

the

Is

absent

predicts

hybridization

indirect

be

calculation

relativistic

the

for

hydrogen

to

between

relativistic

s p i n and

of

RMOs o b t a i n e d

l o c a l i z e d AuH RMOs,

non-relativistic

are

bonding,

cannot

RMOs c a l c u l a t e d

orbitals

of the

β

localized valence

spin

latter

ω .

non-relativistic

σ

This

and

using

changed by

of

au

(18)

predictions

sets,

the

with

the

values

and

e

of

velocity

2.993

cm

using

with β

the

compared

using

become

theory.

of

contracts

calculations

for

basis

vibration

be

substantially

qualitative

The

large

calculations

The p r e d i c t i o n s

from RCI

both

A comparison

experimental

quantitative

AuH a r e

with

to

and

and

value

the

5d

vibration

predictions

respectively,

of

the

b y RMO c a l c u l a t i o n s

increases

3.431

with

resulting ,

the

AFs on

using a

significantly

respectively.

show t h a t

properties

Furthermore,

1

orbital

the

a

various

energies.

increasing

relativistic

values

cm" ,

with

at

only

experimental

several

and fundamental

respectively,

disagreement

2305

of

these

calculation

predicts

the

calculated

these

core,

RCI

wavefunction

addition

relativity

the e

core of

molecular

substantially

a n d u? ,

e

the 20%

was

from

simply by

Thus

e

in

AuH p r e d i c t e d

the

the

corresponding n o n - r e l a t i v i s t i c

that

and

e

frequency cm

predicted

performed

length

of

in

about

the

The bond l e n g t h

then

used)

is

methods

and e x a c t l y

latter

light

placed

energy

placed

Indeed our

an accurate

distances by both

including AFs.

frequency

are

are

set.

which

requires

valence

Au and H atoms.

set

eV,

the

internuclear


chemical basis

computation

configuration

of

the

these

binding

the

associated electrons

enter

295

D

1 0

core

does

in

AuH;

this

electrons set

e

of

calculation with

(EBS), AuH the


and

DF

296


above-mentioned including 2.014

and of

|/?>).

the

We

the

express

|f m >

NRL the

molecular

components

with

terms

the au;

relativistic

total

through

order large

and

3.877

(58916eV!).

defined

|d m >

functions

component

two-component RMOs c a n b e

m >,

than

2165.19

Pyper

m

relativistic

bound at

relativistic

and |p

the

be

RMOs i n where

like

(the

au

3.5

calculated

shall

s

ThO

although

3.877

m

for

molecule

R -

|i m >

eV

molecule

the about

However,

3.5d)

R -

angstrom.

that

at

calculated

experimental

9.0

indicates

predict

NRL D

prediction.

The

0.529171

atoms.

c a n be

calculations

b)

1 au -

experimental

(negative)

au.

non-

separations

Relativistic

)

Relativistic

(ref.

the

18

doubly

(RMO) as

was

and

III.

various

b

ThO

appropriate

DF S C F a n d

various

for

Th atom

Thus,

the of

-0.437,

energies

(D )

dissociation

(au)

a

spinors as

calculated

-0.938,

set

the

orbitals

'cores'

9

Non-Relativistic

R

of

SCF method

taken

relativistic

dissociation

Table

the

atomic

the

The

-1.055, total

basis

thereby

molecular

ThO was

products

-1.246,

in

including

(numerical) for

valence 6d DFAOs

DF S C F w a v e f u n c t i o n .

elsewhere(14).

calculated

and

constructed via

wavefunction

antisymmetrized

the

7s

relativistic

were of

the

thus

6p,

valence

occupied valence spinors

and

6s,

299

equations the

of

the

functions number

momentum

m and correspond

Therefore,

|ρσα>,

have

|ρπβ>,


the |daa>,


300 |άττ0>,

|faa>,

bonding in

|fc70> f u n c t i o n s ,

characteristics

question.

the

double

The v a l e n c e

group

representation has

the

It

-


of

of

|7el/2>

«

|6el/2>

-

|5el/2>

and Th

the

0

(using

the

with

calculated the

D

below

the

small

of

the

|2ρπα>

the

0

claimed

(lying

and

|6dπα>

below

the

8|e3/2>

value

of

D

due

e

au,

energy

about

molecular

corresponding -1.531,

-0.165

au

0.51 D

eV

to

the

2165

energies

at

neglect D

at

R -

-0.837, these

MOs;

NRL c a l c u l a t e d

orbital

energy

for

than

the

ThO a r e

-0.326,

between the

given

lowest

above.

relativistic

lying

valence

Thus

the

IP

of

low

by

-1.50

eV

compared to

the

experimental

u n d e r s t o o d as

the

calculated valence

N R L HOMO ( a t

pure

on the

au)

whereas hybrid with

consists

in of

the

contrast |7sa>

of

an

the

(-0.936)

coefficients

eV p r e d i c t e d

almost

calculated and

given

|βάσ> in

by

the

IP

RHOMO ( g i v e n

parenthesis.

HOMOs i s

NRL c a l c u l a t i o n of

|6d5> o r b i t a l

(-0.350)

the

-0.299

higher

eV.

4.49

for

much

the

The

whereas

3.877

corresponding difference

at

calculated

-0.315,

lie

energies

eV)

NRL

separations.

values

orbital

(16.33

for

eV

total

higher

yield

correlation

and 0.73 the

eV

(18). to

orbitals

easily

the

au

eV

expected

1.81

au

-0.948,-0.887,

0.6

9.00

electron

9 valence of

and

internuclear 3.877

in

whereas

of

eV!)

and

-6.00

au)

not

-1.140,

difference

of

is

for

the

is

IP

8.7

2ρπ 0

ionization eV

of

e

the the

6.14

However,

(58912

these

of

of

reported

R-3.877

between

predict au

involving

orbitals

calculated

(at

lies

e

correponding r e l a t i v i s t i c a

the

respectively.

DF S C F e n e r g y

NRL o r b i t a l

bonding

valence

βάσ

experimentally

NRL c a l c u l a t i o n s is

-0.379|6ρπ£1/2>

DF S C F w a v e f u n c t i o n

and 4.277

can be

and

valence

viz;

Koopman's theorem) is

e

The

3.877

the

just of

substantial

the

effects.

and

RMOs

Moreover,

R -

is

au

forms:

a n d 2s,

experimentally,

There

|8e3/2>

| βάσ>

6daal/2>-0.253|6paal/2>

The

be

Th atom w i t h

12s a n d

2saal/2>-0.123

(IP)

to

(using

irreducible

R =

-0.899 2ρσα1/2>+0.363

agreement

and

at

-0.215

excellent

valence

|9el/2>

-0.Ill|2sal/2>

-0.734|6paal/2>+0.629|2saal/2>

atoms,

the

the

RMO

following

2ρσ and

single

σ - h y b r i d on the

from

a bonding combination of

potential

A

diatomic)

-0.350|6daal/2>

calculated valence

the

valence

(18).

heteronuclear

as

additional

the

species

O.813|2p^l/2>+O.418|6d7r01/2>,

-

6d7r,

the

into

molecular

-0.815|2ρπα3/2>-0.417^πα3/2>+0.095|6ρπα3/2>. three

are

These

is

orbitals

next

RMO)

7s-6d

The v a l e n c e

|8e3/2> The

a

however

valence

for

insight

the

ThO d e s i g n a t e d

notation

the

can give

RMOs o f

form:

contribution

atoms.

RHOMO,

of

which

valence

RHOMO o f

-0.936|7scral/2>

consists

bonding Th

(AIR)

etc,

the

theoretical

following

|9el/2>

of

6.00

above)

orbitals Since

of

the

eV. Th

is

This

7s

Th

DFAO


R

=

atom,

consists the

1.50 too

of atom is

a

21.

MALLI

stabilized is

(due

differences the

in

easily

pure

the

(due the

|2s

the

Th

(due

to

in

the

it

can be

orbital

and

the

considerably

and

that

effects

due

to

atom

the

bonding

in

species

(>2000°C)

in

The

at

5f

electrons in

geometry

(linear)

of

the to

which

For 5f

the

i.e.,

6d a

DFAOs

total

its

core.

was

found

the

of

The at

to

a

the

Th0

is

the

the

the

U atom

U:

the

lowest

au;

diatomic. R -

3.55

the

2346

molecular

total

to

be

2s

the

at

(1

with 3

of

2

the

in

to

R -

3.55

UO p r e d i c t s

relativistic

calculation,

it

=

atom

a

D

Thus, e

of

should not

R -

for

at

of

R

»

6p, 0

7s,

atom;

atomic

[core]

contains

also

was

3.05,

kept

that

a

as

single for

the

calculation

only the

and

and

binding

limit

in

au

s t i l l 4.05

-0.11971

above

twice be

diatomic

-28145.039469

although

almost

This the

the

were

predict

eV)

2 +

2

valence

the

unbound by

27.211

au.

of

mentioned

energy

of

set

the

-0.14393,

fails

the

difference

and

following

where

at

in

the

performed

UO m o l e c u l e

s h o u l d be

UO m o l e c u l e

for

paramount

6d

basis

energy

while

(18). (U)

significant.

DFAOs

0

1

(U0 )

the

the

unbound by

hartree

for

the

were 2p

cm"

orbitals

structure

included

eV)

It

of

and

argon

5f

very

and

eV

in

isoelectronic

a valence

non-relativistic

hartrees energy

Th

of

corresponding non-relativistic

wavefunction the

the

7.8

0

2.5

T h O a n d UO b o n d s .

SCF w a v e f u n c t i o n

au p r e d i c t s

however, about

The

the

actinide

the

however,

be

is 5f

1 6

in

6

to

of

-

e

an

explanation

molecular

respectively.

spinors

6p 5f 6d7s

was

found

e

electronic

electrons

total

R — 3.55

DF

of

of U

roles

were

Γcore]

Is

a n d c*> x

role

and

of

the

wavefunction

4.55

au,

DFAO's

of

and

(-1.68

-0.17596

differ

temperatures

involving

calculations

electrons

viz

the

au u s i n g

au

determinant

6p

high

1

cm"

the

expected

-0.061739

it

energies

the

relativistic

spectrum

atomic

(bent)

2

terms

and U atoms

18

A n IR

825.0

valence

in

at

diatomic

unbound by au

and

of

the

bonding

ThO u s i n g

theories

6d

(-0.969)

DFAO

MO.

d i s s o c i a t i o n energy

eV.

-

β

exist a

Moreover,

molecular and

for

6s

explains

IPs,

significant the

|6sa>

effect)

is

the

the

|lel/2>

orbital

of

of

above

|lel/2>

whereas

predicted

very

investigate

and 4.55 of

configurations, electrons

ω

investigate

4.05

the

RMO

of

DFAO

the

diatomic.

5.6

of

UO m o l e c u l e ,

relativistic 80

to

to

found Th

ThO

of

relativity

3.55,

and

in

between

us

for

predicted

are

claimed

bond.

may b e

orbitals

3.05,

a hybrid

phase w i t h

led

order

prompted in

the

structure

actinide-oxygen

UO

be

the

6d

FOR UO

gas

15°K has

importance in

of

potential

electromic

with

to

relativistic

the

mentioned

valence

stabilization

NRL m o l e c u l a r

UO i s

the

ionization

matrix

atom;

participation

SCF CALCULATIONS The

an

the

orbitals

Th

that

there

the

the

bonding characteristics

relativistic

for

of

energies

stated

and

effect),

lying

The

direct

effect)

relativistic

lowest

calculated

the

Hence

DF

energies The

orbitals.

difference

the

relativistic

indirect

orbital

N R L MO i s

(-0.124)

atom

the

orbital

|6sa> v a l e n c e

and

and

direct

to

understood.

corresponding


to

destabilized

can be

301


-0.973

eV;

UO m o l e c u l e the

the

DF S C F

is

total

NRL

much as

regarded

U0 at

as

the

better

than

DF SCF c a l c u l a t i o n . The

9|el/2>

relativistic consists

from

\5ΐπβ>

Malli

and

and Pyper

of

a

highest

occupied molecular

7sc7-6d m

functions.

s

(where

'e'

representation

ω -

1/2,3/2,5/2),

the

one-electron 9|el/2> -

T h e RHOMO 9 | e l / 2 >

denotes of

the

the

the

total

UO h a s

the

additional

RMO a n d 1 / 2 , 3 / 2 , 5 / 2

where ω i n d i c a t e s RMO),

of

two-dimensional

corresponds

angular

momentum

to

of

viz.

-0.873|7saal/2>

-0.3981 6 o V a l / 2 > + 0 . 1 8 3 | 5 ί π 0 1 / 2 >

- 0 . 1 7 6 | 5 f a a l / 2 > + 0 . 1 2 9 | 6d7r01/2> - 0 . 0 9 3 | 2 s a l / 2 > -0.069|2paal/2>


It

is

an almost

contributions the

0

atom,

However, are

where

the

following

from

n o n b o n d i n g RMO s i n c e

the the

|2saa>

coefficients

RMO 8 | e 3 / 2 >

form which

definitely

lying

clearly

involved

8|e3/2> ~

(-0.093)

in

are

just

shows

the

it

and

very

small

|2ρσα> (-0.069)

contains

DFAOs

given

below that

in

the

RHOMO h a s

|5fcS> a n d

bonding of

UO,

of

parentheses. the

16dn> D F A O s

of

U

viz,

0.727|2ρπα3/2> -0.465|5f*03/2> +0.344|6dπα3/2> -0.111|6ρπα3/2>

The v a l e n c e significant and, has

in

fact,

the

|5ίπα>

and

electrons contain

5f

in

the

DFAOs

the

We g i v e interaction

and

for

ground state

below

and

can o n l y be

destabilizes

these

DFAOs a r e

The

substantial

orbital

roles

the

expressions indicate

2ρπ DFAOs o f

0

clearly

their

for

the

energies 5f

to

in the

(electrons

p u s h e d up

the

associated (which

configuration)

u n d e r s t o o d due

which

spinors.

au)

\5£δβ>,

significantly

effect

very

atom

-0.502

actinides

electronic involved

U

the

results

so t h a t

clearly

the

in

contain the

of

for

(DFAOs)

bonding

6 d DFAOs a r e

energy

-0.38 These

spinors

UO a n d T h O d i a t o m i c s ,

also of

a n d 6 d DFAOs o f

respectively.

results

atomic

indicate

for

8 | e 3 / 2 > RMO a l s o

5f

+0.49

atomic

their

relativistic

UO w h i c h

atom,

-0.74,

5f

these

as v a l e n c e

the

the

significant

ThO,

Both

RMOs a l s o

of

very

6d and 5f

bonding

of

the

DFAO i n

that

indirect act

that

below

from

5 | e 3 / 2 > RMO ( w i t h o r b i t a l

|2ρπα> DFAOs, are

bonding. the)

the

coefficients

demonstrate

except

RMOs l y i n g

contributions

of

in

in

energy

the

and

valence

a n d 6 d DFAOs

in

respectively.

for

the

7|el/2>

and

π-bonding arising and the

due

6dπ a n d / o r

| 5 e 3 / 2 > RMOs to

the

5 ί π DFAOs

of

U

viz, 7|el/2> «

Ο . 7 8 8 | 2 ρ π 0 1 / 2 > +0.34616άπβ1/2>

-0.225 |

5ίπβ1/2>

-0.098|6ρπ01/2>, 5|e3/2> Although,

the

-0.74|5f*j33/2> +0.4915f*a3/2> 6 ρ π DFAO w a s n o t

these

valence

RMOs ( e x c e p t

noted

above);

the

|5f D F A O s

6 ρ σ DFAO o f

contribute

to

found

a very the

to

minor

be

involved

contribution

U contributes 61 e l / 2 )

-0.38 | 2ρπα3/2>. significantly to

a s much a s

RMO w h i c h h a s

the

8|el/2> the

|6da>

following

form: 6|el/2> «

-0.87|2ρσα1/2> +0.30|6daal/2> -0.29|6ρσα1/2> -0.29|5faal/2>

-0.2312s


in

as and

21. M A L L I The

calculated

relativistic

orbital

of the

7|el/2>,

8 | e 3 / 2 > RMOs a n d t h e RHOMO 9 | e l / 2 >

-0.5017,

-0.4551,

-0.4444,

-0.4281 and -0.2266

and u s i n g Koopman's theorem,

the calculated

e V f o r UO i s i n e x c e l l e n t

experimental molecular

value

calculated

agreement

o f 6 ± 0 . 5 eV ( 1 8 ) .

energies,

UO a r e c o l l e c t e d


eigenvalues

6|el/2>,

6.17

303


e t c . , at various

i n Table

IV.

It

IV.

Calculated total

Non-Relativistic

(E

N R

The c a l c u l a t e d

and d i s s o c i a t i o n e n e r g i e s internuclear

seperations

(E

D F

+28144)

( E ^ ) and

( i n e V ) f o r U0 a t (R) i n a u ( l a u -

27.211 eV) various

0.529171

angstrom)

NON-RELATIVISTIC

D (eV)

(E

a

e

M R

+25379)

D (eV) e

3.05

-0.95728

-3.9165

-0.02107

-5.0213

3.447

-1.0394

-1.6800

-0.14147

-1.7451

3.877

-0.98150

-3.2573

-0.12528

-2.1855

4.277

-0.92535

-4.7852

-0.07869

-3.5142

-0.06620

-3.7930

-

4.677

a)

-

The experimental

(negative) respect

value

D

i s reported

e

indicates

t o b e 7.8 eV ( 1 8 ) .

the molecule

to be bound

A positive

(unbound)

Since

the lowest

total

molecular

e n e r g y was c a l c u l a t e d

at R —

a u , t h e c o r r e s p o n d i n g NRL c a l c u l a t i o n was a l s o p e r f o r m e d

3.55

au i n order

3.55

arising

to gain

insight

into

due t o r e l a t i v i t y .

It

eV w h i c h

i s about

energy

i s about

energy

at R -

ionization

2406 h a r t r e e s

3.55 a u .

potential

Moreover,

has

with

the following 9|el/2>NR -

than

that

the t o t a l

at R a D

value.

of

molecular

1.6 eV lower

o f 6 . 1 4 eV w h i c h ,

the experimental

e

NRL m o l e c u l a r

t h e NRHOMO 9 | e l / 2 > p r e d i c t s i s about

at R =

in

predicted by the

t h e DF SCFt o t a l

o f 4.42 eV, which

c o r r e s p o n d i n g DF SCF v a l u e

excellently

was f o u n d t h a t

however,

above

differences

f o r t h e UO m o l e c u l e

0 . 7 0 eV g r e a t e r

c o r r e s p o n d i n g DF SCF c a l c u l a t i o n ;

the

the major

a u , t h e NRL c a l c u l a t i o n p r e d i c t s

-0.97

with

t o t h e two a t o m s .

3.55

bonding

however,

the than

agrees

T h e 9 | e l / 2 > NRHOMO o f UO

expression, v i z . -0.539|5f*01/2>

-0.483|5faal/2>

-0.449|6daal/2>

-0.348|7saal/2>

-0.288|6άπβ1/2>

-0.213|2ρσα1/2>

+0.144|6paal/2> T h e NRHOMO h a s a m u c h l a r g e r 6dπ DFAOs to

contribution

from

the 5 ί π , 5fa,

o f t h e U a t o m a n d t h e 2 ρ σ DFAO o f t h e 0 a t o m

t h e RHOMO; h o w e v e r ,

atom

is

separations.

i n au ( l a u -

RELATIVISTIC R (au)

total

t h e UO m o l e c u l e

internuclear

Relativistic

of

separations f o r

can be seen that

) energies

potential

the corresponding

internuclear

t o be unbound a t a l l these

Table

au, respectively;

ionization

with

5|e3/2>,

o f UO a r e

the contribution

t o t h e NRHOMO i s m u c h s m a l l e r .

o f t h e 7 sa

Moreover,

6d DFAO the

consists mostly

(with

RMO 8 | e l / 2 >

|5f5>

(-0.465)

the

7|el/2>

the

NR 7 | e l / 2 >

(0.381),

has

6d

and

6|el/2>

a coefficient

from

contributions

(0.344)

from

o f t h e U atom.

contributions

from

t h e 2ρπ

DFAOs

above

(0.788),

|5f53/2>

(0.20)

t h e |2ρπ>

Similarly,

o f U; while,

(0.346)

e.g.,

(0.435), 6ρσ

6fa

the

i s a π - t y p e MO w i t h 6dπ

DFAO),

(0.727),

substantially;

o f 2pa (0.746),

( a n d \5fnfil/2>)

RMO a s d i s c u s s e d

and

major

5ίπ

(-0.225)

o f t h e U atom. The

7|el/2>

orbital

about

eigenvalues

a r e -0.4281

respectively, for

DFAOs

a n d |2ρπ>

RMOs a n d NRMOs d i f f e r

consists mostly

(-0.288)

o f 0.90) o f

(0.338)

\5£π>

corresponding DFAOs


(with

contributions

and

i t

4 . 4 eV lower

the 8|e3/2> Our

( t h e NRMOs)

a n d-0.4444

i s clear

than

and

results

o f t h e RMO

(-0.2642)

that

t h e NRL o r b i t a l

7|el/2>

therefore

clearly

demonstrate

predictions

o f t h e NRL a n d D F S C F c a l c u l a t i o n s

etc.,

as quantitative

energies,

f o r the diatomics

energies

MOs.

as well

total

eigenvalues are

t h e c o r r e s p o n d i n g DF SCF o r b i t a l

qualitative bonding,

8|e3/2> and

(-0.2989) a . u .

orbital

energies,

involving

that

differences

there

f o r the nature

dissociation

actinides

a r e marked

between t h e

duet o v e r y

relativistic

effects

i n such

RELATIVISTIC

EFFECTS

FOR D I P O L E MOMENTS O F D I A T O M I C S

of

energies significant

systems. O F HEAVY

ELEMENTS The

RIPhas

for

the diatomic

using

been

ab i n i t i o

chemical

basis

internuclear value

basis and

a u was

dipole

functions,

Moreover,

dipole

relativistic Pyper are

(14).

predicted

dipole

experimental

and

PbTe,

about

values

respectively,

turns

40%

out that

au differ

and

0.019

au,predicted with

TiH,

+

a n d NRL

and

27 ( E B 2 7 ) (STO)

f o r AuH ( 2 1 ) .

EB27) a n d

reported

and B i H ;

a u and

by Malli and dipole

moments

however, t h e

very

1.0623

dipole

well

with

au f o r T i l

moment

f o r

the

+

moment

calculated

wavefunction s e tused

the predicted

(at R

e

=

i n the calculation

dipole

s e twavefunctions) from

thevalues

b y t h e NRL w a v e f u n c t i o n s , of TiH, (CB

f o r AuH

i s smaller (by

b y t h e c o r r e s p o n d i n g NRL

thebasis

In the case +

a

In

asA B " .

predicted

considerably

v i z Ti H";

f o r which

f o r AuH u s i n g t h e and

au agree

a positive

the relativistic

polarity

+

orbital

reported

experimental

PbH

1.2655

chemical basis

respectively.

expected

where

PbH

distance.

20 (EB20)

s e t (EB20

(NRL)

t h e CB s e t r e l a t i v i s t i c

However,

0.371

been

BiH,

limit

a t the experimental

except

calculated

(18) o f 1.8137

that

and

calculated

(with a u and

depending upon

relativistic

BiH,

+

the dipole

the

have

moments and

the relativistic with

at present, HgH ,

PbTe

b y a 6p S l a t e r - t y p e

were

the relativistic

the wavefunction.

from sets

basis

i t s polarity

t o 50%) t h a n

wavefunction, of

moments

au) using

species

(CI) wavefunctions

f o r AuH,

AB i n d i c a t e s

It 2.8794

curves

extended

o f 1.9078

the

basis

b y a 6p DFAO,

Unfortunately,

wavefunctions)

species

calculated

interaction

notavailable

PbH ,

(WF) c a l c u l a t e d

e

extended

dipole +

T i l ,

as n o n - r e l a t i v i s t i c

( R ) o f each

moment

CB s e t ,

configuration

as well

CB s e t a u g m e n t e d

CB s e t a u g m e n t e d

TiH,

used f o r the internuclear

moments

using

(21) t o evaluate +

HgH ,

s e twavefunctions

separation

obtained

b y Ramos

s p e c i e s AuH, relativistic

o f 3.5884

addition, WFs,

adapted

although

moments

o f 1.372, f o r AuH,

of dipole

0.323 -0.120

T i H and

the dipole

set)wavefunction

the value

(using

o f 0.976,

moment

moment

predicts the (-0.12 au)


21.

MALLI

obtained

from

polarity

for

moments AuH, (/i

),

>

A*CB

EB27

Ε

>

^CB


is

at

au,

(μ

Ε Β

R

=

e

2.8794

by

the

moments

(21)

electron

Table

R

C

I

n

c

the *

internuclear W

^ R C I ^

that

at μ

ε

RCI

that

and

the

diatomic in

study

systems

Tables

effects

than of

Ε

Β

2

it

Ε

in β

2

fairly

set

A*

C B

and

lesser μ

polarity

the >

7

0.967

order: μ

Ζ Β

and

a

that

predicted

lesser

polarity both

by

chemical basis effect

can be

that

>

7

for CB

order:

that

the

dipole

the

of


and

V-VI

are

au,

relativistic

first

μ

a

are

R C I

thereby

the

in

0.967

that

3.3794

wavefunctions

au,

are

R C I

the

relativistic

^ F S decrease

and / *

β

opposite

separations

decrease

thereby R «

ε

from

3.3794

(MRCI) μ

s

and / z

at

the

F

R =

β

the

A comparison of

predicted

MS c a l c u l a t i o n s ,

presented such

of

indicating

is

correlation of

*

limit

of

DF S C F LCAS

moments

(

au,

RCI

This

set

indicating a

respectively,

dipole

results

N

au,

27)

non-relativistic

using

A

^ R C I> except

wavefunctions. on

2.8794

set

predicted

the

=

e

>

^EB27

0.846

(MEB2 7^

a

Ti"H .

values

μ

respectively,

EB27

whereas

for

the

+

viz

^RCI » except

R

au,

) ,

Β

at

that

set >

^EB27

0.846

c o r r e s p o n d i n g N R L WF i n d i c a t e s molecule

calculated

(μ),

whereas ( μ

the this

indicates

C B

305


set

relativity

atoms

or

ions

concluded from

relativistic

significant

our

and

for

dipole

systems.

V.

Dipole

Calculated

Moments

for

(μ)

LiH,

Til

by U s i n g Chemical Basis

b

and

PbTe

Wavefunctions

8

d

c

A-B

it ( a u )

LiH

2.575

Til

1.908

1.814

PbTe

1.266

1.062

a

Reproduced with

+

A B"

c

polarity.

reference

(18).

l e

EXP (2.367)

p e r m i s s i o n from au -

2.542

d

D.

U s i n g extended

Ref.

2.314

e

b

21.

A l l

Experimental

basis

values

values

function

of

indicate

from

Malli

and

Pyper

(14).

CONCLUDING We h a v e LCAS

REMARKS

c o n c l u s i v e l y shown f r o m

MS

calculations

quantitative diatomics

features

that of

properly

understood using

based

on

the

that,

for

Schrôdinger

the

chemistry

the

core

the

6p

to are

the

in

and

sixth of

gold

heavier 6p

row

DFAOs for

and v e r y the

not

In the

relativistic

well

as

(Z

>

Moreover, in

of

90)

in

cannot

non-relativistic and

it

it

is

theory

heavier

has

5d DFAOs

gold

safe

DF S C F

the

and bonding

atoms

compounds, whereas

involved

chemistry

as

addition, 5d

non-relativistic

the

fully

structure heavy

equation.

and mercury

are

initio

traditional

elements,

elements.

semi-empirical

significant

ab

qualitative

electronic


the

the

be

theory

been

shown

participate they

to

chemistry

in

belong

state (in

predictions),

to

that contrast but

elements.


they

306


Table

VI.

AuH R e l a t i v i s t i c

Dipole

Non-Relativistic


Values

at

Moment

Dipole

R ,

c

in

e

Curves,

(μ)

Moment

au

8

and

b

d

Wave f u n e t i o n s CB R(au)

+ CB

2. 6294 2.. 8 7 9 4

+

6DSL

f

6pDF

-

0.968 g

CB e

-

RCI

EB20

EB27

0.905

0.829

0.738 0.802

0.976

1.065

1.271

0.984

0.901

(1.373)

(1.443)

(1.925)

(1.434)

(1.363)

3. . 1 2 9 4

0.967

-

-

1.046

0.956

0.846

3. . 3 7 9 4

0.968

1.132

1.045

0.909

8

-

C a l c u l a t e d by u s i n g the

Chemical

Basis

functions,

(CB),

EB20

functions),

-

wavefunctions

extended b a s i s

(same

as

EB27 b u t

and r e l a t i v i s t i c b

wavefunction.

Non-relativistic

without values

polarization

6 p DFAO a s

(see

reference

We a l s o 5d,

structure

DFAOs

involved

relativistic

significant

in

hoped that

would fully

our

(and to

in

the

of

a u -

(RCI)

parentheses.

2.542

D.

centered

effects) the

the

prediction

of

for

the

the

knotty

gold

in

e.g.,

the

electronic

third-row the

formidable

for

dipole of

a

bottlenecks

systems. be

moment,

fairly and

properties

relativistic of

DF S C F c a l c u l a t i o n s h a v e b e e n b r o k e n ,

dual

of

such

shown t o

non-energetic

quality

that

are

calculation

effects

CB atom

18).

heavy

have been

f

on g o l d .

for

present

accurate

effects

All

Slater-type

DF SCF c a l c u l a t i o n s

and they

the

properties,

Thus,

l

in

associated electrons)

correlation

criterion

future.

d

6p'

(reference

e

initio

and a c t i n i d e s ,

accurate

R

diatomics

relativistic

relativistic

ab

of

given

2.75

relativistic

and e l e c t r o n

the

-

polarization

interaction

21.

ζ

(14);

polarization

MCDF c a l c u l a t i o n

their

the

non-energetic

supplement

wavefunction

the

quantum c h e m i s t s

Furthermore, is

(due

elements to

exponent

from

conclude from

5 f

are

CB plus

Experimental

and bonding o f

transition challenge

9

14).

e

polarity.

obtained

6d and 5f

definitely

the

Au H"

function with

plus

the

+

indicate

reference

including

configuration

R e p r o d u c e d w i t h p e r m i s s i o n from Ref. values

from

EB27

ab and


initio it

is

it

21.

MALLI


gratifying for

that

performing

calculations

the computational reliable

for diatomics

It

i s hoped that

ab

initio

with

machinery fully

fully

c o n t a i n i n g heavy

is currently

relativistic

c o n t a i n i n g heavy

the a v a i l a b i l i t y

(all-electron)

polyatomics near

ab i n i t i o

and very

of faster

relativistic

atoms w i l l

hand

DF SCF heavy

atoms.

supercomputers

calculations

become

at

307

feasible

for i n the

future.

ACKNOWLEDGMENTS


I

sincerely

inviting to

Yu, My for

Professors Dennis symposium.

Salahub

T h i s work

and Mike

Zerner

for

h a s b e e n made p o s s i b l e d u e

t h e c o o p e r a t i o n a n d t h e e n t h u s i a s m o f my c o l l e a g u e s a n d c o w o r k e r s

over to

thank

me t o t h i s

many y e a r s ;

in particular,

Dr. N.C. Pyper, for their thanks their

Research financial

also

D r . R.

contributions go t o

cordial

would

cooperation. through

like

t o a c k n o w l e d g e my d e b t

M e s s r s A . F . Ramos a n d D.

to the research reported

the operations

C o u n c i l o f Canada support

I

Arratia-Perez,

no.

is

in this

paper.

o f our Computing S e r v i c e s

The N a t u r a l

(NSERC) grant

staff

Sciences and Engineering

thanked

for their

continuous

A3598.

LITERATURE CITED 1. Schrödinger, Ε. Ann. Physik. 1926, 81, 109. 2. Klein, O. Z. Physik. 1926, 37, 895. 3. Gordon, W. Z. Physik. 1926, 40, 117. 4. Dirac, P. A. M. Proc. Roy. Soc. Lond. 1928, A117. 610. 5. Burrau, O. Kgl. Danske. Videnskab. Mat. Fys. 1927, 7, 14. 6. Dirac, P. A. M. Proc. Roy. Soc. Lond. 1928, A123. 714-33. 7. Mayers, D. F. Proc. Roy. Soc. Lond. 1957, A241. 93. 8. Swirles, B. Proc. Roy. Soc. Lond. 1935, A152. 625-49. 9. Boyd, R. G.; Larson, A. C.; Waber, J. T. Phys. Rev. 1963, 129, 1629-30. 10. Mingos, D. M. P. Phil. Trans. Roy. Soc. Lond. 1982, A308, 7583. 11. Hay, P. J.; Wadt, W. R.; Kahn, L. R.; Bobrowicz, F. W. J. Chem. Phys. 1978, 69, 984. 12. Ziegler, T.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys. 1981, 74, 1271. 13. Jiang, Y.; Alarez, S.; Hoffmann, R. Inorg. Chem. 1985, 24, 749-57. 14. Malli, G. L.; Pyper, N. C. Proc. Roy. Soc. Lond. 1986, A407. 377404. 15. Grant, I. P.; Mckenzie, B. J.; Norrington, P. H.; Mayers, D. F.; Pyper, N. C. Comput. Phys. Commun. 1980, 21, 207. 16. Brown, G. E.; Ravenhall, D. G. Proc. Roy. Soc. Lond. 1951, A208. 552-9. 17. Sucher, J. Phys. Rev. 1980, A22, 348-62. 18. Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules: Van Nostrand Reinhold, New York, 1979. 19. Pyykkö, P. Chem. Rev. 1988, 88, 563. 20. Lee, Y. S.; McLean, A. D. J. Chem. Phys. 1982, 76, 735.


THE C H A L L E N G E OF d AND f ELECTRONS

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21. 22. 23.


24. 25. 26. 27. 28. 29. 30.

Ramos, A. F.; Pyper, N. C.; Malli, G. L. Phys. Rev. 1988, A 38 2729-2739. Katz, J. J.; Seaborg, G. T.; Morss, L. R. The Chemistry of the Actinide Elements. Chapman and Hall: London, 1986. Oetting, F. L.; Rand, M. H.; Ackermann, R. J. The Chemical Thermodynamics of Actinide Elements and Compounds Part 1: International Atomic Energy Agency: Vienna, 1976. Oetting, F. L.; Fuger, J. The Chemical Thermodynamics of Actinide Elements and Compounds Part 2: International Atomic Energy Agency: Vienna, 1976. Erdos, P.; Robinson, J. M. The Physics of Actinide Compounds: Plenum Press: New York, 1983. Handbook on the Physics and Chemistry of the Actinides Vols. 15; Freeman, A. J.; Lander, G. H. Eds.; North Holland: Amsterdam, 1987. Grant, I. P.; McKenzie, B. J.; Norrington, P. H.; Mayers, D. F.; Pyper, N. C. Computer Phys. Commun. 1980, 21, 218. Ackermann, R. J.; Rauh, E. G. Higher Temp. Sci. 1973, 5, 463; J. Chem. Phys. 1974, 60, 2266. Hildenbrand, D. L.; Murad, E. J. Chem. Phys. 1974, 61, 1232. Malli, G. L.; Oreg, J. J. Chem. Phys. 1975, 63, 830-841.

RECEIVED March 21, 1989


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