Correlation energy contribution to cohesion in covalent structures

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Correlation energy contribution to cohesion in covalent

structures

Peter Fulde, Gernot Stollhoff

To cite this version:

Correlation energy contribution

to

cohesion in covalent

structures

Peter Fulde and Gernot Stollhoff

Max-Planck-Institut für

_{Festkörperforschung,}

D-7000

_{Stuttgart 80,}

F.R.G.

(Reçu

le 6 mars 1989,

_accepté

le 5 mai

1989)

Résumé. 2014 La contribution des corrélations

_{électroniques}

à

_l’énergie

de cohésion des structures covalentes est discutée. Par ce _{moyen la discussion} est faite entre les calculs ab initio sur une base finie d’orbitales de _type

_gaussien

et les calculs

_simplifiés

de corrélation. En considérant le

diamant, le silicium et le

_{polyéthylène}

en détail, on montre _{que les différentes contributions à}

l’énergie

de corrélation ont une

_{signification physique simple.}

Abstract. 2014 The contribution of electron correlations to

the cohesive energy of covalent structures is discussed.

_Thereby

a distinction is made between ab initio calculations done within a

finite basis set of

Gauss-type

orbitals and

_simplified

correlation calculations.

_{By considering}

diamond, silicon and

_polyethylene

in detail it is shown that the different correlation contributions have

_{simple physical meanings.}

Classification

Physics

Abstracts

71.10 - 71.45G - 71.45N

1. Introduction.

The

_concept

_{of the chemical bond has proven} to be a

_simple,

_yet

_highly

successful scheme for

describing many molecules and

solids,

in

_particular

those formed

_by

elements on the

_right

of the first row of the

_periodic

table. For more details the well known textbooks of

_Pauling

_[1]

and Harrison

_[2]

should be consulted. An

_important

_quantity

which one would like to

calculate and to _{understand in detail is the cohesive energy. For}

_solids,

i.e.

_periodic

lattices,

refined total energy calculations based on the Local

_Density

_{Approximation (LDA)}

to the

Density

Functional

_theory

have been

_performed

_[3].

However,

often

_{simplified computational}

schemes are

_desirable,

in

_particular

when one wants to elucidate trends and overall features. For that purpose Friedel [4, 5] gave a

_{simple analysis}

of cohesion in covalent structures, based

on a Hückel

_{approximation. Thereby}

he noticed and

_pointed

out the

_importance

of electron correlations for a

_{quantitative understanding}

of cohesion. He gave estimates of their

contributions

_[5].

Since then this

_{important point}

has obtained considerable attention and calculations with

_quantum

_{chemical accuracy have become available} even for solids. Stated

differently,

correlation energy contributions to cohesion in solids can be calculated with an

accuracy common for small

_systems,

like

_CH4

_(methane).

The aim of this paper is to discuss those ab initio

_type

of calculations of the correlation contributions to cohesion. From them

_parameters

can be

extracted,

which may be used as

input

for

_simplified

models.

The framework within which electron correlations are treated is that of the Local Ansatz

[6],

for which

_quantum

chemical _accuracy was demonstrated before. The way in which this

theory

is

_presented

here differs from

_previous

ones. Here the relevant

_equations

are derived

by using

the

_operator

_{(or Liouville-)}

_{space instead of the Hilbert space. For}

_systems

with

weakly

correlated

electrons,

like in most covalent

_bonds,

this new formulation leads

_only

to a

rederivation of

_{previous equations.}

But it has the

_advantage

_{that it may be also used for}

strongly

correlated electronic

_systems

_{[7]. Therefore it may}

serve as a

_step

towards a

_unifying

description

of weak and

_strong

correlations.

The paper is

organized

as follows. In the next section the

_equations

are summarized which are

_required

for ab initio calculations of cohesive

_{energies. Applications}

of the formalism are

found in section 3 where a number of

_examples

are discussed.

_{Simplified computational}

schemes follow in section 4. In that section also a

_comparison

with the ab initio results is

provided

and an extraction of model

_parameters

is described. Section 5 contains a _summary

and the conclusions.

2.

_{Computational}

method.

The ab initio calculations of the correlation energy are done within a finite basis set

fi(r)

of

_Gauss-type

orbitals

_(GTO’s).

In terms of them the Hamiltonian is written

The

operators

_{ai’ aju}

fulfill the commutation relations

where

is the

_overlap

matrix of the basis-set functions. The Hamiltonian is divided into a self-consistent field

_(SCF)

_part

_HSCF

and a residual interaction

_part

_Hres,

i.e.

Let

1«/10)

denote the

_ground

state _{of H with energy}

_Eo

_and

_{1 cPSCF)}

the

_ground

state of

HSCF

with _energy

_ESCF.

The correlation _{energy contribution}

_Ecorr

to the

_ground-state

energy is

It can be

_expressed

in a

_simple

form when one works in

_operator

-

or Liouville _{space with} a

metric [7]

properties

are introduced and discussed in detail in reference

_[8].

From reference

_[7]

it follows that the correlation energy can be written in the form

The

element

,f2)

in Liouville space

can be considered as

_representing

the correlated

_ground

_state

_{1 tPo),}

in the same _way as the wave

operator à

_does,

which transforms the SCF

_ground

state 1 CPSCF)

into the exact

_ground

state

1 tPo),

i.e.

The

_quantity

_Lo

in

_{equation (8)}

is a Liouvillean or

_{superoperator}

which acts on other

operators A according

to

Any

approximation

in il reflects itself in one for

_Ec.,.

Because

_Ecorr

is

_{completely expressed}

in terms of

_cumulants,

it remains size consistent when

_{approximations}

for Q are made. This

means that the correlation energy is

_always

_proportional

to the electron number when the latter becomes

_large.

As is well

_known,

some of the standard

_quantum

chemical methods like

the

_{configuration}

interaction

_(CI)

method do not fulfill this

_requirement.

In order to

_compute

_{Ecorr approximations}

for 1Ii are

_required.

One

_{possible approximation}

is to limit all calculations to an essential

_part

_ao

of Liouville space and to

_neglect

comple-tely

the

_{complementary}

_part

_:RI

_[9].

Assume that

_fllo

is

_{spanned by}

the elements

JA,,)

( v = 1, ... M ).

Then

_equation

₍₇₎

reduces to

where P is a

_projector

which

_projects

onto

_{fllo. By applying}

_{projection techniques}

it is a

_simple

matter to show

_that

This

_equation

can be considered as a

_{special coupled-cluster approximation,}

or

_{alternatively,}

as a

_generalized

form of a

_coupled

electron

_{pair approximation}

_(CEPA-O).

The second

_{important approximation}

concerns the form of the

_operators

_Ai.

For them the

Local Ansatz is used

_[6].

It takes into account the local character of the correlation hole and

thereby

reduces

_decisively

the

_complexity

of the calculations.

_{Starting point}

is a set of local functions

For a

_given

i the

_{f m (r)}

are limited to GTO’s centered at a

_particular

atom. The

_operators

bi (bi )

create

_(destroy)

electrons in local orbitals

_{gi (r)}

with

_{spin .}

With their

_help

and

_spin

_operators

can be defined. The three

_components

of a are the Pauli matrices. From them the

_following

local

_operators

are constructed

The elements in Liouville space

with

describe local

_two-particle

excitations out

_{of 10 SCF>}

but in a _{way which} ensures size

consistency.

The Local Ansatz identifies the

_elements

_{J A,)}

with

_the

_150,,).

The correlation energy follows then from

equation

(12).

3.

_{Applications.}

In the

_following

a number of

_applications

of the above formalism are discussed.

_They

deal

with the determination of the correlation contribution to _{the cohesive energy of diamond}

[10, 11],

silicon

_[12]

and

_polyethylene

_[13].

The

_computations

were

_{performed by using}

a finite basis set

_consisting

of

_Gauss-type

orbitals

_(GTO’s).

For the SCF

part

of the calculations a double-zeta basis set was chosen in all three cases. For diamond and

_polyethylene

it consisted of four

_(contracted)

s orbitals and two sets of

_{(contracted) p}

orbitals as far as the C atom is concerned. For the H atom the basis consisted of two s orbitals. In the case of Si a

_{pseudopotential}

was used for the core electrons

[14]

and a basis

_consisting

of three

(contracted) s

orbitals and two sets of

_{(contracted) p}

orbitals for the valence electrons.

_Very

_recently

the calculations for diamond and silicon were

repeated by including

a set of d

functions,

i.e.

_polarization

functions,

in the above calculations

[15].

This was made

_{possible by}

_making

use of the program

_package

CRYSTAL

_[16].

The

SCF

_{binding energies}

are obtained

_{by subtracting}

the

_{corresponding}

atomic

_energies

calculated within the same

_{approximation.}

The deficiencies in the former can be estimated

from calculations for small molecules

_{using larger}

basis sets

_[17].

The final results are listed in

table I

_together

with the estimated uncertainties.

_They

refer to the limit of a

_complete

basis

set, i.e. the Hartree-Fock

_(HF)

limit. For further details the

_original

literature should be consulted

_{[10-13, 17].}

As

_regards

correlation

_energies,

a distinction is made between interatomic and intra-atomic

contributions. Interatomic correlations suppress fluctuations in the valence electron number between different atoms.

_They

can be described within a minimal basis set, i.e. one which has as _{many basis functions} as there are valence electrons. In that case the local functions

gi (r) (see

Eq.

(13))

are the atomic

_parts

of the bond

orbitals,

after

_they

have been

Table 1. - Valence-shell correlation and

binding

energy contributions

for

diamond,

silicon and

polyethylene

in _{eV per unit} cell.

_Binding

_energies

are obtained

_by

_adding

(1)

and

(2)

and

subtracting

the

_free

atom values

_(3).

atoms

_they

are

_essentially

sp3 hybrids

with small

_{orthogonalization}

tails on

_neighboring

sites.

On H atoms

_{the gi}

_(r)

coincide with the atomic s

_orbitals,

_again

with small

_{hybridization}

tails.

Once the

_{gi (r)}

are determined the

_Om

_operators

follow from

_equation

_(16),

and the correlation energy is obtained from

_equation

_(12).

The results are

_presented

in table I. The contributions of interatomic correlations to _{the total correlation energy} are of the order of 20

to 30

_percent.

_They

contribute

_(almost)

_fully

to _{the cohesive energy (for} more details see

Refs.

_[6,

11,

13]).

The dominant

_part

comes from the

_operators

_{ni 1 ni 1 . These}

_operators

and

operators

of the

_{form ni nj}

where i

_{and j}

denote

_hybrids

on the same atom reduce fluctuations in the valence-electron number at an atomic site.

_By

means of the

_operators

_si

_sj

_(see

Eq.

(15))

Hund’s rule correlations are introduced.

_Operators

of the

_{form ni nj}

and si

_sj

with i

and j referring

to

_hybrids

on different atoms allow to include correlations over

larger

distances which turn out to be van der Waals interactions between bonds.

The

_largest

contribution to _{the correlation energy}

_originates

from intra-atomic correlations.

They

describe correlations between electrons on the same atom and hence the

_strong

_spatial

variation of the correlation hole of an electron at small

distances,

i.e. within an atomic volume. Due to _{the cusp in the}

_pair

distribution function in the limit of small distances

(correlation cusp),

the intra-atomic correlation energy is

only slowly

convergent

with

increasing

size of the basis set. Since this

_problem

occurs for a

_single

atom as well as for a

molecule or

_solid,

the contribution of intra-atomic correlations to cohesion is much less sensitive to the size of the basis set. For a reasonable

_description

of intra-atomic correlations

the use of

_polarization

functions is

_mandantory

(i.e.

one set of d functions in the case of C and

Si atoms, and one set _{of p functions in the} case of H

atoms).

The functions

gi (r)

which are

decrease.

_They

subdivide the atomic volume and need not be

_orthogonal

with

_respect

to each other. Their detailed

_description

can be found in reference

[11-13].

The results of the

calculations are found in table I. Also listed are

_neighboring

site and

_overlap

corrections.

They

are small and are obtained

_{by treating}

_{simultaneously}

intra-atomic correlations on

neighboring

sites.

This is the most elaborate

_part

of the correlation calculations. It does not involve more

numerical work than

_treating

a small

molecule,

like

ethane,

within the same basis set.

_By

present

standards this is a rather trivial

task,

indeed.

There are a number of

_shortcomings

_{of the way in which} correlations have been treated. One is the Local Ansatz itself. It misses

_typically

of the order of 5 % _{of the correlation energy} which can be obtained within a

_given

basis set. A second one is the limited size of the basis set

discussed above. From

_computations

on small molecules with C or Si atoms and

_comparable

bonds it is known that deficiencies caused

_by

the basis set are

_roughly

twice as

_large

when an

atom is

_part

of a molecule or solid than for a free atom

_[17].

The estimated corrections of the

two

_shortcomings

to the correlation

_energies

are added to the final values in table I. One notices that electron correlations contribute between 21 % and 24 % to cohesion in the three

cases under consideration. The results obtained set a standard with

_respect

to _{accuracy of}

correlation calculations for solids and

_polymers.

Recently

a calculation of the correlation energy for the

_ground

state of diamond was done

by using

a Jastrow wave function

_[18].

This

_{implies working}

with

_specific

_two-particle

functions. The basis set

_problem

_{is avoided this way.}

_Using

a Jastrow wave function

corresponds

to

_assuming

a correlation hole for electrons which is translational invariant. The

computation

of the

_expectation

value of the

_ground

state _energy was done with the

_help

of Monte _{Carlo calculations. The correlation energy obtained within these calculations agree} well _{with the correlation energy listed in table 1}

_despite

the fact that the

_long-range

part

of the correlation hole is assumed to fall off

_{exponentially}

instead of

r- 6

as

_{required by}

the van der

Waals nature of correlations.

It _{is very instructive} to consider the

_{pair-correlation}

function in more detail for the

_systems

under consideration. As will be shown it differs

_considerably

from the one taken from a

homogeneous

electron gas calculation. The dominant

changes

in the

_pair

correlation function

come from interatomic

correlations,

in

particular

from

_operators

of the form ni t ni j and

ni

_{n ..}

The

changes

decrease fast with

_increasing

distance r due to the van der Waals nature of the interatomic correlations at

_large

distances. Therefore the relevant

_quantities

to

_study

are

and

The

_{subscripts i, j}

refer to functions _{gi (r)} used for the

_description

of interatomic

correlations,

and furthermore to

_pairs

of states on the same atom. For

_{nearly homopolar}

bonds

(ni t ni ) =

0.25 and

_(nz

_{nj> ;z=}

1. Listed in table II are

_Aji

and

_Liij

for the different bonds in

CH4,

C2H6

[17],

Si2H6

[12]

and the three

systems

diamond

_[11],

silicon

_[12]

and

_polyethylene

[13],

respectively.

It is noticed that the individual correlation corrections compare well with each other.

_Therefore,

the correlation hole remains almost

_unchanged

when the size of the

system

increases. This is in accordance with

_simple

intuition. In distinction to what one

_might

Table II.

_{- Changes}

in the

_{pair-correlation}

_function

caused

_by

electron correlation. For the notation see

_equations

(19, 20).

The

_subscripts

a and b

_refer

to

_hybrids

on C or Si atoms

pointing

in the direction

_of

_(a)

H atoms and

_(b)

C or Si atoms,

_{respectively.}

Finally,

a

_comparison

is made of the total

_charge

fluctuations on a C or Si atom in different chemical environments. For

_{nearly homopolar}

bonds it is

where n denotes the

_operator

of the total valence electron number on atom A. Values for

(dnÃ)SCF

and

_{(An A 2 )c.,r}

with A

_referring

to a C or Si atom are

_given

in table III for different

systems.

It is seen that the reduction of total

_charge

fluctuations due to correlations is almost the

same in the different

_systems

under consideration.

Table III.

_{- Charge}

_fluctuations

on the individual C and Si atoms.

4.

Simplified

correlation calculations.

The correlation calculations can be done

_analytically

when one restricts onself to a minimal basis set, and when the Bond Orbital

_{Approximation}

_{(BOA) [2]}

is made. Consider elemental semiconductors. In that case the

operators

a:’

create an electron in an atomic

sp3

_hybrid.

The SCF

_ground

state in BOA is of the form

The two

sp3

_{hybrids forming}

bond I are indexed

_by

the

_subscripts

1 and

2,

respectively.

In BOA the

_{hybridization}

_{par Ho} of the Hamiltonian

simplifies

to

The interaction

_part

of the Hamiltonian is

_{approximated by}

The

_part

_Hl

includes the electron interactions at a

_given

atomic site A. It is here reduced to

three Slater

_parameters.

One,

Ù,

is a Hubbard like on-site Coulomb interaction while the

second,

JI

is an on-site

_exchange

interaction. The third

_parameter,

i.e.

_J2,

enables one to

distinguish

between the Coulomb

_repulsion

of electrons in the same and in different

sp3

hybrids.

Thus

Terms which are not

_biquadratic

in f

and

f’

do not contribute to

_ground

state correlations when the BOA is made. Therefore

_they

have been

_dropped

here. The index

_{f(= 1,}

_{...4 )}

refers to the different

_{sp3 hybrids}

on atom A.

The Hamiltonian

_H2

contains the Coulomb interactions of electrons on different sites

A,

A’. It is

The Hamiltonian used here differs

_slightly

from the one used in reference

[19]

which included more interaction matrix elements. The

_advantage

of the

_present

form is that its relation to a

multiband Hubbard Hamiltonian is better seen.

In BOA the

_density

matrix takes the

_simple

form

Because a minimal basis set is

used,

only

interatomic correlations can be calculated.

Correlations within a bond have a similar effect like in a

_H2

_molecule,

i.e. those

_{configurations}

are favoured in which there is one electron at each of the atomic site

_{participating}

in the bond. Correlations between different bonds are

_{significant only}

when the bonds involve

hybrids

of

essential

_part

_{of Liouville space}

_ao.

_They

are

_{given by equation}

(17)

with

_Om

of the form

(compare

with

_Eq.

_(16)),

The

_long

_range

_part

of the van der Waals correlations discussed in the

_preceding

section is

neglected by

this choice of the

_{Om. When the correlation energy is}

determined

_according

to

équation

(12)

it is found that the

_only

_part

of

_H2

which contributes are matrix elements

VAA,,

where the

_hybrids

_(A,

_l)

and

_{(A’ , 1 ’ )}

_belong

to the same or nearest

_neighbor

bonds.

They

are denoted

_by

_V1,

_V2

and

_V3

and from

_figure

1 one can read off which

_hybrids

are

involved in the three cases. It turns out that

_they

appear in the correlation energy calculation

only

in the combinations

_{( U}

+

_{2 J2 -}

V 1 )

and

( U -

2

_{V 2}

+

V 3 ),

_{respectively.}

Therefore it is

helpful

to introduce renormalized interactions

_Ueff

and

_Jeff

_through

Fig.

1. - Notation used for the Coulomb matrix elements between different

_hybrids.

When in

_equation

₍₂₇₎

the

_quantities

_(!7+2./2)

and Ù

are

_{replaced by}

_{(Ueff + 2 Jeff)}

and

Ueff ,

then with

_respect

to _{correlation energy calculations the effect of H2 is} included and therefore that

part

of the Hamiltonian need not to be considered further. When

Ueff

is written as

U - V

_{1 +}

[- 2 V 2 + V 1 + V 31

one can

_interpret

the sum of matrix elements

in brackets as a

_polarization

contribution which increases

_{(0 -}

_Vi).

Since

_only

two kinds of correlation

_operators

are

_included,

one is left with two

_independent

_parameters,

which enter

into the correlation calculation for the

_ground

state. The

_parameters

in the Hamiltonian

Hl

condense into

The fact that

_only

two

_independent

_parameters

_{appear in the} correlation calculations can be

One notices that in terms of

LI

and

J the

_parameters

_{ql, q2} are of the form

_{q1 to =}

"0 + 2 J and

When the interatomic correlation energy is calculated

according

to

_{equation (12)}

one finds

In order to determine the

_parameters

_{ql, qz for diamond and silicon} one can

_proceed

in two

different ways. One consists in

extracting

the values of the different matrix elements from an

ab initio SCF calculation of the

_type

discussed in the last section. That was the

_approach

used in reference

_[19].

For

_example,

for diamond the

_following

values were

_found,

(/+2./2

=

22.5 eV

_(note

that this

_{quantity corresponds}

to U in Ref.

_[19]),

_JI

= 1.9

eV,

and

V

= 13.3 eV. One notices that

(U

+ 2

_{J )}

is

_considerably

less than

_{( Û}

+ 2

_{J 2)}

because of the

large

intersite Coulomb interaction

_Vl.

Here we

_{proceed differently,}

i.e.

_{by making}

use of

and

_{by taking}

those values from the ab initio calculation described in the

_preceding

section. This _{way the following} values are found for diamond

_(C.)

and silicon

_(Si.)

When the values for diamond are

_compared

with the

_previous

ones, one notices a difference in

_{( LI + 2 J )}

_by

1 eV. The source of this difference are

_improvements

in the ab initio

calculations

_[11]

as

_compared

with reference

[10, 19].

The fact

that J (C,,,,)

= 1.5 eV is smaller

than

_Yi(Coo)

= 1.9 eV

reemphasizes

the

_strong

renormalization due to

_{polarization.}

For

_comparison

the values

_{( U}

+ 2

_{J2 ), J2}

and

_Jl,

were also determined for a

_single

C and Si

atom from an ab initio calculation. The

_following

values were found

A

_comparison

with the values for diamond shows

strong

renormalization effects. For the

atoms it holds

_{that J2 :::. Jl.}

The relative small value of

_{J (C. )}

indicates that

Jeff

is

_appreciably

smaller

_{than Jz.}

This is due to a sizeable contribution of

[-

2

V 2

+

_V

₊

_V3]

as

_explained

in

the text

_{following equation}

_(31).

In order to determine the

_parameters

_{ql, q2} one needs to know also the value of

to.

It can be obtained

_by

a fit of the bands obtained from a BOA with those obtained from an

Hartree calculation

_(see

Ref.

_[19]).

The values

_to(Coo)

= 10.7 eV and

to(Si,,)

= 5.0 eV _were

the

_following

_{values for the interatomic correlation energy per unit cell} are

_obtained,

E inter(C.

= 2.3

_[eV/uc]

and

_E,,ter(Si.)

= 1.1

[eV/uc],

respectively. They

_are in

fairly good

agreement

with those of the ab initio calculation

_(see

Tab.

_I).

When

_only

the

_operators

₍₃₀₎

are used in such a calculation the

_{corresponding}

values are 2.2

_[eV/uc]

for diamond and 1.1

_[eV/uc]

for

_silicon,

_{respectively.}

But when the

_long

_range-part

of the van der Waals correlations as well as

_spin

correlations are

_included,

and furthermore when the deficiencies

of the Local Ansatz are corrected for the

_following

total interatomic correlation

_energies

are

found

These numbers differ somewhat from earlier ones in reference

[19].

The reason for this

difference are

_improved

values for the interaction matrix elements from ab initio calculations.

The

present

results of the

_simplified

calculations

_reproduce

well the exact ones

_given

in table I.

The same has been found to hold true for a considerable number of small molecules

involving

first row atoms

_{[20, 21].}

Until now

_only

interatomic correlations were

_considered,

because of the minimal basis set

which has been used. Intra-atomic correlations can be accounted for

_{by using}

an « atoms in molecules »

_type

of

_approach

_{[22, 20].}

This is

_possible

because the two

_types

of correlations

are almost

_independent

of each other. Within that

_approach

the intra-atomic correlation energy per unit cell is

given by

Here

_{P corr (v)}

is the

_probability

of

_finding

at a C atom

_(in

the case of

_diamond)

or a Si atom

(in

the case of

silicon) v

valence electrons. The

_{corresponding}

atomic correlation energy is

£;orr.

The latter can be

_decomposed

into

where Wv

(i )

denotes the relative

weights

of different

_{configurations}

when v valence electrons

are

_present

and

Eat(i)

is the correlation energy of v valence electrons in

_{configuration}

i in a free atom

_(ion).

The latter can be found in the

_literature,

in tabulated

_form,

for a

number of atoms of the first row

[23, 24].

What is

_required

is a

_{specification}

of

PCOff(V)

and

_{w v (i ).}

For

_pcorr(v)

a Gaussian distribution of the form

is chosen centered at

_4,

_{which is the average valence electron} number in elemental semiconductors. Its width is determined

_by

the total

_charge

fluctuations

_(An’).,,

_(see

Eq.

(21)),

and can be

_computed

like in the ab initio case discussed before. For the relative

weights

w"

(i )

it is assumed that

they

are the same as _e.g. on the C atom in

_CH4,

so that

_they

are

_easily

determined.

for a free C atom, one obtains the contribution to cohesion. Instead of

_doing

_this,

a

_simple

algebraic

expression

of the form

can _{be used for the intra-atomic correlation energy of} a first-row atom A

_{[20, 21].}

It was

obtained

_{by investigating}

and

_analyzing

a

large

number of molecules

_consisting

of atoms of the first row.

_Thereby

it was found that the final result for the intra-atomic correlation _energy of an atom A

_{depends only}

on _{its average number of valence electrons}

_nA

and on the ratio

rP =

np /iiA

of the p electron

_{number n-P}

to

_fiA.

_{Equation (43)}

contains also the correlation contributions of the core electrons and therefore should not be

_{directly compared}

with the

corresponding quantities

in table I. For the intra-atomic correlation of H atoms in molecules a

similar

_{simple algebraic expression}

is

_found,

_namely

In the

_following

_equations

₍₃₉₎

and

₍₄₃₎

are

_applied

to

_compute

the correlation _energy contribution to the _{cohesive energy of diamond. For} an evaluation of

_equation

₍₄₃₎

the values

fie

= 4

and rc

= 0.75 _are used. This

yields

the

following

intra-atomic correlation

energy per unit cell

When the free atom value of

_{E(C)=20134.26[eV]}

is subtracted two times

_following

contribution of intra-atomic correlations to cohesion is found

Together

with the interatomic contribution

₍₃₉₎

this

_gives

a total contribution of correlations

to cohesion

This value agrees

reasonably

well with the ab initio value listed in table I.

A similar

_analysis

has been done for

_{polyethylene.}

The value obtained for

_{EBorr agrees}

within 5 % with the one of the ab initio calculation found in table I.

5.

Summary

and conclusions.

It _{has been demonstrated that correlation energy} calculations for covalent solids can be done on an ab initio level. With their

_help

one can determine the correlation contribution to

cohesion. It was also shown that the ab initio results can be well

_{approximated by}

a

_simplified

model calculation based on the BOA. The model Hamiltonian contains on-site interactions

only,

with renormalized matrix elements. In

_particular

the on-site Coulomb interaction is

strongly

renormalized. This is due to a considerable interaction of electrons on

_neighboring

sites,

which is

_incorporated

in the renormalization. The

_theory

has been

_applied

to

_diamond,

silicon and

_{polyethylene.}

In the three cases correlations contribute between 20 % and 25 % to

cohesion.

Acknowledgements.

We are

grateful

to the

Quantum

Chemistry

group at the

_University

of

_Torino,

in

_particular

to

Dr. M. Causa for

_making

_{their program CRYSTAL available} to us. We also would like to

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