Simple atomic models

327

doumalofMolecultzrS~ture,12O(1986)327-336 THEOCHEM E&e&r

Science Publishers’ B_V., Amsterdam

SIKPLE

ATOMIC

M.

-

Printed in The Netherlands

MODELS

BERRONDO

lnstituto

de

Ffsica,

Cuernavaca,

Her.,

U.N.A.M..

Aptdo

Postal

139-B,

H&xico

62190,

ABSTRACT We present a comparison between different models of atomic structure, based on functionals of the density and simple model potentials, as well as en improvement of the minimal ST0 basis.

INTRODUCTION

1.

The vels

simplest

for

view,

the

this

atomic

model

is

non-relativistic

is

far

from

Bohr

model,

hydrogen

atom.

trivial1

The

one

which From

gives the

dimensional

the

correct

semiclassical

energy point

quantization

leof

condition

(1.1) becomes

exact

points

can

comes

exact

when

where

is

the

equation.

original

to

of

contour,

For

variable2

radial The

in the

r=ex

quantum

only so

integration

infinity

asymptotically.

change

nr

original

be deformed

Langer’s

dial

the

the

result

the

the

number,

(1.2)

complex

Coulomb

gives

singularities

contour

case,

surrounding plane,

radial

counting in

this

is

exact,

since

we have

the

turning

WKB theory

a radial

be-

equation,

and

condition:

the

number

integrand and

of

are

gives

nodes

of

contained

the

the

ra-

inside

quantization

the con-

dition:

(1.3)

Taking If

Bohr’s

angular

we now turn

one-electron

our

quantum attention

number to

k as the

k= x+1,

radial

it

Schradinger

atom:

016S1280/86/$03.30 01985

starts

ElsevierSciencePublishers B.V.

at

k=l.

equation

for

the

323

(1.4) we see The

that

It

has

behaviour

at

two

regular

these

two

singular

points

at

the

origin

and

at

infinity.

points:

(1.5)

gives

the

exact

nodeless

Matching

solutions.

the

asymptotic

conditions:

(1.6)

determines

their

states

be understood

can

tion.

This

example

ferential re

the

namely,

the having

are

Fock

model 4y5.

due

Its

the

and

lent

starting

serts

that

also

valid

as well

for as

of

more

the

singularities

general

cusp-like

of

is

oroital

conditions

field

of

net

attraction,

in

the

atomic

and a-@ for

dealing

virial

into

account.

in

an of

excited condiof

the

difwhe-

equations3, at

the

origin,be-

effects

This Thus

becomes the

6

case,

spin

also

The

exchange the

potential case:

with

theory term

in

antiparallel

important acting

on

the

in

orbital

number 9

in

asstate,

variational

Hartree-Fock

the

ground

imbalance

between

accounts

correlation

when

an exceltheorem

the

as is

porepul-

variational

constitute

perturbation-like

Hartree-Fock

while

the

per

resulting

an average

Brillouin’s

interact

Hartree-

spin-orbital

The

fulfills

principle.

the

one

and

orbitals

do not in

exclusion

function.

calculations5.

particularly

exchange

antiparallel

.

It

ingredients,

excellence:

attraction

Hartree-Fock

both in

wave

nuclear

configurations

drawback

Pauli’s

par

We associate

N-particle

sophisticated

two essential

and

known.

the

The

simplification

spin

well

are

model

interaction.

excitation

parallel

the

with

more

correlation

large.

are

theorem.

for

a great

the

virtues

interelectronic

the

there

contained

The main

well

atoms,

a superposition

single

this

many-electron

main

point

calculations.

very

both

hence

to

principle

is

of

instead is

vanishes

the

is

a central

indeed

electron,

d-a

importance

shows

condition,

case

They

tential

in

important.

For

and

This

The nodal behaviour for the n =O). r terms of the (weighed) orthogonality

(with

simply

clearly

equation. asymptotic

come very

sion

eigenvalues

is of

not

very taken

electrons

Hartree-Fock

N

Shell bitals

models and

such

assumes

and the X, model 7 starts

Hartree-Fock

as

each

that

electron

moves in an average

An alternative

approach

is

to view

varying

density,

as

in the Thomas-Fermi

to

charge

include

the charge This

exchange

consists

density,

taking

expression

the place

of

tential

( and modifications

in the orbitals’

In any event, of

dealing

inclusion

of

garding

Orbital

density

results Slater

exclusively,

for

of

view,

the

orbitals

as a functional

theory, to give

of

point.

used7 since

in

this

a non-local

po-

in which for

wave functions

too awkward,

restrictive,

inclusion

of

self-consistent,

while

(with

particularly

constraints

the

the use of re-

imposed by

this

without

depends

potential

3 contains of

such as Hartree-Fock

potential

the orbital?.

improvement (STO)

are

way

particles. into

section

to be the most satisfactory

to be too

effective

equations while

regarding

out

N-particle

instance)

and the proper

number of

and models

has proven

proves

the one-particle

linear

turns

10

modification

as the starting

in Hartree-Fock

in practice.

can be devided

to

type

energy

finite

models

matrix,

rise

mer point

is a prescribed In section

the

;tlinimal

latter.

basis

increasing

on the density

which

the number of

or

function,

2 we present

Section

sets,

and

gi-

the

4 includes modifies

the

forsome

the

parameters.

DENSITY FUNCTlONALS The total

matrix. matrix higher

energy

of

the atom can be calculated

In the Hartree-Fock is enough

y (1,2)

to the fact order

that

the

density

approximation, to calculate

density

matrices

matrix

are

all is

knowledge the physical

assumed

expressed

from the second

as

to

be

of

the first quantities.

idempotent.

(antisymmetrized)

variation of the total energy with respect one. In fact, 11,12 matrix to the two constraints of idempotency: , subject

the first sity

factors

in which

ving

2.

correlation

and

gas

field. a slowly

) have been extensively

and thus

orbitals

structure

the kinetic

energy

or-

equations.

atomic

density

antisymmetry

matrix

the use of

with

the charge

X &,

term appearing

on the density

electron

thereof

gas with

model 8.9 . Dirac’s

the exchange

the homogeneous

the exchange

term depends

electrostatic

the atom as an electron

in writing

from electron

order

density

order

density

This

is due

Hence all products

of

to the den-

(2.1) and

330

(2.2) gives

the correct Since

equations.

the kinetic

energy

charge

density

does

bitals

is very

important

formation

for

sity. for

even

the ground

This the

if

problem

does

term here

part

of

sity. of

is

it

is

The remaining

field

approximated density

In the atomic

thesis

the orin-

to be monotonically term

in the Thomas-

is very

picture,

attraction

which,

where

far

from satisof

the den-

the expression

contribution

as functionals the kinetic

is

repulsion,

The dominant given

in terms

a substantial

by an average

account

in the Coulomb part, numerical

different. indeed,

interelectronic

pieces

well

of

and this

on the gradient

is entirely

and correlation

Their

the

one.

the

included

form for

for

charge

den-

the correction

and the deviation

from an

is small

can be

of

the density.

energy

in the case

of

and they

a slowly

varying

by:

case

function in this

nuclear

out energy

terms depending in the orbital

structure energy,

the density

exchange

‘T-

creasing

turns

produced

reasonably

is given

The nodal

by the Coulomb repulsion

behaviour.

The functional

of

operator,

the kinetic

The kinetic

the situation

Regarding

the self-interaction

average

of

which

atoms.

the correct

energy,

given

of

correction

is

density.

purpose.

is a functional

the attractive

the charge

this

density,

not appear

energy

For the potential

of

state

which

we include

kinetic

for

by a differential

in the computation

in the charge

Fermi epproximation, factory

represented

not suffice

is erased

decreasing

is

CrcW

however, whose

case

is

=

the

(2.3) radial

logarithm to evaluate

y c,, 2)

c

density

is almost T hW+

behaves as a monotonically 13,14 . A more realistic

linear

assusIng

a density

matrix

of

dehypo-

the form

‘I[%) (2.4)

where %(rl is almost

1 inear.

The kinetic

energy

then

turns

out

to be ‘5:

(2.5)

331

written

in terms of

The expression

the density

(2.5)

is

ken as a correction

precisely

to the

the one found

free

electron

gas

16

by Weis&icker

Eq.(2.3)

choosing

.

It can be ta-

a different

coeffi-

cient:

(2.7) as suggested for

by Ki rzhnits

al 1 the atoms,

trends

17

the main value

and the asymptotic

Let

us next

expression

look

obtained

. Since

at

these of

behaviour’ the exchange

from the electron

approximations

these for

expressions

large

energy gas

are

N,

is

rather

not very

to obtain

accurate global

than specific

as a functional 10 is :

of

cases.

the density.

The

(2.8) with&

=l.

This

The proper and Kohn

term overestimates

value E

the variation

of

a

turns

the exchange

out

effects for neutral atoms 18 to be intermediate between the original

18-20

.

&=I,

Sham’s value 19 &=2/3, and decreases with N. A justification for 18,21 22,23 of d with N can be given by modelling the Fermi hole .

The correlation

function

can be approximated

as:

(2.9) ,yielding

a value

of atwhich

depends

on the number of&

electrons

N,

as:

(2.10) A different use Eq.(2.4)

starting for

point

the atomic

in the calculation

density

matrix,

of

the exchange

energy

is

to

with (2.11)

where

‘I(r)

sults

in:

is almost

constant

14

. The crudest

approximation

in this

case

re-

(2.12) while

the correction

depends

on the density

gradient,

since

(2.13)

(2.14) yielding

a fairly

accurate

expression

for

the exchange

energy

14

.

ATOMIC POTENTIALS.

3.

The original tion It

of

average

the non-local

is given

exchange

potential

appearing

as a multiplicative

24,25

potential

was

introduced

in the Hartree-Fock

as a simplificaorbital

equations.

operator:

(3.1) with 64,

and is

determined

The modification fulfill

proposed

the virial

orbitals.

in Ref.18

chooses

in the calculation 7 method , the potential

procedure,

yielding

in the Hartree-Fock-Slater

the value

theorem

In the X,

self-consistent

20

self-consistently

of

of

ti in such a way as to

the energy

(3.1)

is

spin-orbitals

model.

from the

resulting

used as a spin-unrestricted,

which

are

quite

close

to the

UHF orbitals. In this is

picture

incorrect.

nentially,

of

behaving

as can be readily

the case

of

negative

mogeneous

electron

t ree-Fock

does.

A local Eqs.(2.4) as well

the asymptotic

however,

Instead

ions,

where

13

potential

(2.11).

It depends

as on the density

p(G)

-t/r

behaviour for

local

which

This

exchange

does

explicitly

and its

of

large

in Eq.(3.1). do not give

gas expression

exchange and

seen

as

the exchange

distances, is

particularly

potentials

26

gradient

through

in

from the ho-

malady

uses

to the nucleus

y(q),

expo-

, even when Har-

from this

on the distance

potential

decays

dramatic

computed

a bound state

not suffer

it

r,

Eq.(2.13):

(3.2) where ;)(?\

= 2+1 (3.3)

This it

potential

should.

is

finite

at

the origin,

and decays

as

-Vr

asymptotically,

as

833 A further for

simplification

Coulomb repulsion

ming an exponential tial

turns

out

results

and for

by taking

the exchange

form for

a -bona fide potentia1*7’*3 in the orbital equations.

part

the density,

the Coulomb repulsion

both Assu-

effective

poten-

to be28:

(3.4) where

% is a parameter from Eqs.(3.2)

lows

determined

and (3.3)

29 . The exchange

variationally

taking

7 also

as a variational

potential

fol-

parameter:

(3.51 Orbitals

calculated

the unoccupied

scheme, case,

but

form of ions

with

refer

this

orbitals

to excited

are

potential states 29 .

The effective

very

not virtual

orbitals

the exchange

and excited

29 give

potential

created

(3.5)

good total

orbitais,

by the presence

is also

energies.

as

of

the approximate

In this

in Hartree-Fock’s N-l

electrons.

one for

ihe

negative

potential:

(3.6) behaves

as -Z/r

at

the origin,

stood

as a Coulomb-like

tween

the nuclear

at

large

4.

potential

charge

with

asymptotically.

a screened

2 and the asymptotic

charge

charge

It can be under-

which

Z-N+1

felt

interpolates by the

be-

electron

distances.

MINIMAL BASIS I:artree-Fock

theory

can be compared. Nevertheless, tion,

both

orbitals

in this

for

bottleneck zeta

basis

sets

sets

the correct

no longer

of

distances, of

of

Slater

atomic

zeta while

Hartree-Fock results,

potential

is

type

basis

the use of

approach

no longer

atomic

models

a tumbling

(algebraic)

these

atomic

to Hartree-Fock, stone.

equations.

The The most

are the minimal basis and the double 30 . orbitals Indeed the exponential behaviour as shown in Eq.(l.5).

the

the smaller

a minimal

as for

the

other

calculation

Hartree-Fock

basis,

of

which

results become a standard. basis 30 gives much informa-

In the algebraic

one asymptotically, for

with

in a fixed

the non-linearity

in atomic

an integer

In the case

numerical

the solutions

the exchange

using

In the double small

the yardstick

calculations.

then becomes

basis

sense, of

interpretation

in molecular

popular

constitutes

expansion

the non-locality

is

and (-Z+N-1)/r

larger one

The power

s however,

orbitals. exponent is

the exponent

closer turns

gives to out

the dominant

J-xc

term at

3 as in Eq.(1.6).

to be some average

value

is

between

the two zeta

We can

values

introduce

of

the double

an optimized symptotically,

Eq.fl.6).

origin

wer expansion cusp

to

the

gives

condition.

for

the

logarithmic

basis. by Jetting

so that

we modify

the correct

Thus,

basis

minmal

have as an exponential Close

zeta

its

exponent

behaviour

in

derivative

a Is wave function

the basis tends such

at

function

be-

to behave

a way that

the origin,

as

the

as

in po-

in the

we can write

&5 its> which

behaves

=

\

(4.1)

as:

(4.2) Hence the parameter

a can be taken

z-g

Q”

while

f

is determined

The rest at

of

the origin

(4.3)

variationally

the s functions

is given

as

as are

by Eq.(h.l),

in the minimal

taken once

as normal

basis

STO’s,

the orbital

is

with since

built

STO’s. the

behaviour

as a linear

combi-

nation. For higher

values

of

4,

we use a similar

procedure,

so that

(4.4)

with

(4.5) 0 As a result the minimal larly

for

we have a minimal

STO. The total

the

otherwise

lighter

basis

energies

atoms.

This

are is

with

the same number of

almost

illustrated

of

double for

zeta

parameters

quality,

a few cases

as

particu-

in Table

I_

in

335

TABLE Atom

ST0

He

fulfills near

oz

ours

2.847

Be

With

1

2.8615

14.557

2.8617

14.568

14.5723

Ne

127.81

128.396

128.535

Mg

198.86

199.468

199.607

Ar

525.76

526.370

526.815

the basis

defined

the proper combination.

ved by asking

cusp

above,

because

condition,

The resultsare

the orbital

we do not guarantee

itself

nevertheless to fulfill

of

that

the mixture

the orbital resulting

very

encouraging 31 the condition

itself from the

li-

and can be impro-

(4.6) CONCLUSIONS

5.

As a conclusion viour

of

gross

features

the orbitals

In the bly

we would

both of

at

simply

the origin

the effective

like

to

recall

the

and asymptotically, potential

intermediate region, all we need 29 showing some shell rtructure.

as of

is a smooth

importance since

it

the orbitals interpolation

of

the beha-

determines

the

themselves. scheme,

possi-

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