Multiconfigurational calculations using Elementary Jacobi Rotations

357

JoumalofMoleculmStructure,12O(1985)857-363 7!HEoCKE~ EkevierSciencePubli&eraB.V., Amsterdam-Pr&ediuTheNetherlands

MULTICONFIGURATIONAL

CALCULATIONS

USING

ELEMENTARY

JACOBI

ROTA-

TIONS

R. CARBOl, 1 Secci6 de Institut 2

Dept. Av.

Ll.

DOMINGO1

Quimica Quimic

Quimica

and

Quantica, de

647,

J.

NOVOA'

Dept.

Sarrifi,

Fisica,

Diagonal

J.

of

Chemometrics,

08017-Barcelona,

Fat.

Quimica,

SPAIN.

Univ.

08028-Barcelona,

Barcelona,

SPAIN.

ABSTRACT A variational method based on Elementary Jacobi Rotations is described, applications on mono and multiconfigurational energy expressions are implemented and some examples given.

INTRODUCTION Unitary

transformations

of MO

obtain

optimal

multiconfigurational

widely

used

recent

et

(ref.

al. MC

form

in

1) for

theoretical of the

although

Following tions we

(ref.

present

ples

in

previous 4)

order

scheme.

so

succesive

ponential

and

In

to

can

advantages

in

the

following

a)

The

theoretical

usually

be

mentioned

the

order has

to

been

review

EJR

can

of EJR

based

an

of Robb

exponential

(g = exp(z),

and

(refs.

3).

extension and

be

some

Jacobi expressed

will

result

2 and

to MC

xf-lr)

computa-

problems

computational Rotations in in

exam-

(EJR)

exponential an

(ref.5)

equivalent

on

procedure

points: becomes

simpler.

may

be

a

form, ex-

of M0l.s.

an elementary

formalism

on

on monoconfigurational

Elementary

transformation such

instance,

procedure

implement

of

in

wavefunctions

matrix

based

a detailed

applications

The

are

theoretical

each

coefficients

details.

work

the

fact,

global

for

transformation

exceotions

CI (MC)

See,

developments

here

MC

more

unitary

some

times.

and

resumed

368

b)

Exact

optimization

order

at each

cl

Convergence elements

to

of

pairs

can

be

performed

up

to

fourth .

step. the

dealing

d) Control

of MO

the

optimal

with path

MO

set

a couple

towards

is

of

the

divided

orbitals

energy

into at

a sequence

each

extremum

of

time.

is

completely

assured.

THEORETICAL The

DEVELOPMENT

energy

expression

in

a general

MC

framework

can

be written

as:

(1-a) or

Eb

=gz (w h + tf 1 j ij ij

where

E

= E

functions, a {hij\ over

MO's

sity

matrix

EJR

the

set

of polyelectronic

CI variational

coefficients

((ij/kl)')

are

the

and

while

{wij\,

usual

igijkl)

one-

are

the

and,

two-electron

first-

and

state in

(1-b)

integrals

second-order

den-

elements.

is possible

chosen. such

is

{OI\

(1-b)

)

the

coefficients, been

(l-a)

(ij/kl)

are

r$

and

It

; in

rijkl

to

optimize

depending Each

in

on which

variational

equation

(1)

form

or

goal

can

(Ea be

either Eb)

achieved

the

of the

CI or MO energy

by means

of

For

ii,>

=

an

as:

a) For CI coefficients , = CI cos(x) - CJ sin(x) cI C' = c I sin(x) + CJ cos(x) J

b)

has

MO

(2-a)

coefficients

ii) cos(x)

-

lj) sin(x)

Jjl} .= Ii) sin(x)

+

lj) cos(x)

A transformation

of this

kind,

(2-b)

cam

be

represented

by

a unitary

359

matrix

such

as

cos(x)

g=

sin(x)

Using

(3)

formulae

variation

produced

Variation lynomial

by

(2.b)

energy

possible

is

be

of

expressions sine

polynomial can

it

a transformation

rotation

sines

nomials a)

the

order

rotation

or

of the

in

fourth

(2-a)

in

when

obtained.

(1)

From The

compute

this

in

these

the

energy

type.

gives

applied

(1.b).

to

a second

case

order

(l.a),

and

polynomials,

corresponding

po-

a

optimal

variational

poly-

are:

CI

=

AEa

(c-l)

Ez

+

s Eo"; +

cs Et:

(4-a)

+ s2 Eo"; ; c+s')*

b) MO

AE~ =

-E:: +

(l-c)

EM0 04

s4

. J

where

the

cated

expressions See

and

CI

each

indicates,

as

When ble

to

is the the

fficients.

The

made.

If

ch

least

at

it was one

are

most

the

are

(Ez+

E~c)s~

+

(E$}

integrals

In

at the is

are and

compli CI

the

first

one,

same

time,

while

optimized

other.

reached

(4-b)

coe-

details.

possible.

set

in

in

in

the

search,

in

constant,

Computational our

the

experience

that

the

first

stable.

exact

a MO

more

of the

have

sine

next

and

transformed

space

as we

optimal

compute

+

electronic

5) for

coefficient

optimized, far

(4.a)

on the

4 and

coefficients one

in

schemes

previously

procedure

(Ez\

(ref.

implementation

second

(Eti+Eyi~)s~

(4-b)

depending

Two

the

i

c=(l-s2)3

coefficients

fficients.

MO

(E:~+E~~c)s

+

step,

of

a given

energy

of the

increment

depends

rotation, rotated

rotation and

on what

then

the

indices

is

type

known,

the

are

is possi-

rotated

of rotation

molecular appear

new

it

integrals computed

coe-

has

been

in whiagain

360

using

the

-a small

expressions

subset

fficients

are

CI rotation and

second

density

matrix

and

CI

in the

CONVERGENCE The on the the

their curve, atom. space, dy

in

those are

relative

relative

values

on

defined order

to

Clementi

the

possible

the

the

obtain

If

a

first

the

new

coupling

no

between

truncation

normal

has

shape

of

of the

curve

was

subsets ST0

or Eo4<0

dealing

In with

[slr

been

droped

b)

correspond

to

the

same

behaviour

active

more

than

circumstance,

polynomial

were

there

(4-b)

the

out

last

minimum

for

simplicity.

already

one

is

noticed

specific

is possible,

with

Cases for

of MC a high

the

and mixtu-

MO's. care

of this

algorithm

must

be

the

sine

range

sine

one

used

stu-

1

has

that

in the

CI

sc 0

=j

and

any

Helium

dimensional

exponents

6).

para-

important,

the

The

MO

case,

for

six

of

variation

chosen

cases.

(ref.

when

of the

more

energy

made

form

of those

orbitals,were

possible

Roetti

and,

the

entirely

the

value

range

the

depends

define

any

principle,

their

atomic all

process that

superindex

whole

when

the

the

To take

of the

In

form

cases

monoconfigurational

of

to

coe-

out.

compute

because

parameters

Eo4< 0 j, - IEO,I

re

exact

of

configuration

and

c)

implies

polynomial

carried

them

only

5).

is

defining

include

Eo4” Eo2

and

is

that

expressions.

find

by three

b) Eo2<0,

a)

the

is to

important

curves. To

possible

IEoJZ

not

of the

values

a search

a) Eo2> 0,

where

is

increment

three

above,

step using

characteristics

is possible.

of

rotation

next

(ref.

variational

convergence

All

the

Then

means

PROPERTIES

energy

meters

computed).

matrices,

described

4 (this

reference

another

elements

coefficients

made

is

and

order

process

been

evaluated

performed,

The

in

integrals

was

Hamiltonian

MO

of

derived

process

able

three to

find

is made in

situations,

order

the

at the to

the

absolute

minimum.

begining

locate

if

sine

of the

there

optimization X search

in

optimization

is more

than

onu

361

TABLE

I

Total

energy,

iterations

lA. I

----_ 3B

1

for

dipole the

moment CR2

EJE

(both

in

a.u.)

and

number

of

computation

MONOCONFIGDI?ATIONAL: . . . . . . . . -38.848867 Energy Dipole moment . 0.87977 9 Iterations .... MULTICONFIGURATIONAL: Energy . . . . . . . . -38.874969 Dipole moment . 0.89686 14 Iterations .. .. CI space .... .. 11 configurations and 12 Active CI MO's. 4, occupied: (lb2)(3al) virtual : (2b2)(4al)

sf;ate

functions

--------------------__________________I_~~_______ MONOCONFIGURATIONAL: Energy . . . . . . . . -38.909564 Dipole moment . 0.24513 7 Iterations .... MULTICONFIGURATIONAL: rA: Energy . . . . . . . . -38.912747 Dipole moment . 0.25017 6 Iterations ... . .. .... CI space 4 configurations and 4 state Active CI MO's_ 4, occupied: (lbl)(3al) virtual : (2bl)(4al)

functions

l B:

. . . . . . . . -38.924254 Energy Dipole moment . 0.22758 10 Iterations .... .... .. CI space 8 configurations and 12 state Active CI MO's_ 4, occupied: ~;>~~;a,~ virtual : 2 al lc: Energy . . . . . . . . -38.929238 Dipole moment . 0.23453 Iterations . . . . 39 . . . . . . 20 configurations CI space and 28 state Active CI MO's. 6,occupied; (lb2)(3al)(lbl) virtual: (2b2)(4al)(2bl)

functions

functions

662 minimum

and,

Using

if

the

algorithms

Helium

atom

of the

configurations 3-7

(only

construct other be

always

the

to

is the

absolute

discussed

converge

to

if

the

same

final

The

suitable

one.

above,

the

selected.

iterations)

cases,

due

which

so,

whole

MO

subsets

space,

for

the

number

of global

iterations

the

MO-CI

coupling

CH2

structure

the

fact

that

point,

process

CI

example

computations

all

double

independently

ie

are

on the

very

chosen

fast

in

order

excitations. increases

becomes

and

to

In this

can

important.

RESULTS Some tion

example.

GTO-MC and

calculations

program

PDP-VAX

of Dunning tained

sures slow

named

implemented ARIADNE The

7),

and

reference

8.

(ref.

starting

via

have

versions.

from

The ned

We

on

the

vectors

same

a good

the

in

formalism

starting

on

outlined

has

UNIVAC

System

basis

set

MC

was

was the

procedure

as

an

above 80,

found

experimental

level. may

a

4341/2 work one

a monoconfigurational choices

in

in the

previously

starting

applica-

IBM

were

Other

point.

here

ideas

geometry

the

given

the

with

chosen

are

ob-

obtaiThis

lead

en-

to

convergence.

The Other

results systems,

obtained are

are

resumed

currently

in

studied

Table

and

I for

will

the

CH2

molecule

be published

else-

support

the

where.

ACKNOWLEDGMENTS One "Caixa The calona

of us

(JJN)

dlEstalvis authors

acknowledges de

have

computational

Barcelona" benefied

of

the

finantial

and

the

Sperry

British Rand

and

from

Council. University

of

Bar-

facilities.

REFERENCES 1 M- A. Robb and R. H. A. Eade, (I. G. Csizmadia and R. Daudel ed.) D. Reidel. Dordrecht, 1981, pp. 1. 2 H. Hinze and E. Yurtsever, J. Chem. Phys., z (1979) 3188. 3 T. Chang and F. Grein, ibid., 57 (1972) 5270. 4 R. Carbb, Ll. Domingo and J. JTPeris, Adv. Quantum Chem., 15 (1982) 215.

363 5 R. Carb6, Ll. Domingo, J. J. Peris and J. J. Novoa, J. Molec. struct., 93 (1983) 15. 6 E. Clemenz and C. Roetti, Atomic Data and Nuclear Data Tables, Academic Press, New York, 197'4. Theoretical Chemistry, vol. 7 T. H. Dunning and P. J. Hay, Modern Plenum Press, New York, 1977, pp. 1. 8 G. Herzberg and J. W. C. Johns, J. Chem. Phys., 54 (1971) 2276.

3