Potential energy function of polyatomic molecules

Potential Automatic

Energy Function

Determination

of Polyatomic

of the Unitary Transformation

for the Perturbation Treatment

f.ahoratoire

de Spectroscopic

Molecules Operator

of the Hamiltonian

.IfoE@rrrtaire, 1’nicersiiP dr Furis

I -I, P-1IZIS 5he, France

AND %.

(‘1HI.A

Institute oj Plrysical Clrerr&r_v, Czechnslouak A caderry oj Srimces,

l’ra,q, Czechoslor’akiu

;\n original method for the automatic determination of the unitary transformation operator that occurs in the Van Vleck’s perturbation method is presented for the case of the vibrational Hamiltonian of linear triatomic molecules. This method is a general one and may be applied to the vibro-rotational Hamiltonian of more complicated molecules even when one or several rotational resonances occur simultaneously with one or several vibrational resonances. Numerical results are given for the case of CO;! (vibrational Hamiltonian). ‘The computation time is discussed.

In the first paper of this series (I), we presented a brief outline of an original method for the accurate determination of the coefficients appearing in the expansion, with respect to mass-independent internal coordinates, of the potential energy function of a pol!.atomic molecule, with application to the carbon dioxide molecule, up to the fourth order of approximation. This function was applied to the computation of the vibrational energy level structure of nine isotopic species of this molecule. The calculation gave very good agreement with the observed vibrational energy values. More specific information about the method we used has been published recently (Z,3). However we have not \-et discussed the problems raised by the use of successive unitary transformations of the Hamiltonian to solve corresponding parts of the SchrGdinger equation. This is the first purpose of this paper, the second being to present a generalization of this fruitful perturbation method. Such generalization will lead us to introduce a new notation scheme (see Section III of this paper). For the sake of simplicit\- we shall restrict ourselves to the cibration Hamiltonian of a polyatomic molecule.

It is well known that the SchrGdinger equation corresponding to the Hamiltonian operator for a rotating-vibrating molecule does not lend itself to an exact solution. It is also well known that this Hamiltonian may be expanded in orders of magnitude with respect to the normal coordinates of the molecule. This expansion, which depends upon

475

CHEDIN

416 these coordinates

and their conjugate

AND CIHLA

momenta,

may be written

as

H = Ho + AHI + . . . + XW,.

(1)

Here, h is an expansion parameter taken equal to unity, the exponent of which indicates the maximum order of magnitude of the contribution to the energy of each of the terms appearing in (1). Thus Ho >> HI >> . . . >> H4. The principal term HO of this espansion is the sum of the Hamiltonians of a certain number (depending, in part, on the number of nuclei) of nondegenerate or degenerate harmonic oscillators. If it is assumed that the zeroth-order Schrodinger equation Ho#o = Eo$o

(2)

has been solved, it is possible to proceed with the perturbation calculations. The contributions to the energy due to HI, Hz, . . . may be evaluated by the usual methods of the perturbation theory in quantum mechanics. However, it is immediately seen that this is a formidable undertaking, especially since the zeroth-order energies may be degenerate. This is the reason which has led many authors, and originally W. H. Shaffer, H. H. Nielsen, and J. H. Thomas (I), to use another method, discovered by J. H. Van Vleck (5), based upon some particular transformations of the Hamiltonian into other operators H’, Ht, . . ., respectively called once, twice, . . transformed Hamiltonians. These operators have the same eigenvalues as the untransformed Hamiltonian but are much more amenable to perturbation treatments. It can readily be shown that they are related through canonical transformations (also called “contact transformations”) of type H’ = THT-’ (3a) H’ = W’F’, where T and 7‘ are unitary

operators

. .,

conveniently

(3b)

chosen as (6)

T = exp(iXS},

(-la)

T = esp{ iA%}

(4b)

where S and S are Hermitian operators. If we replace, in Eqs. (3), H, H’, and Hi b> their respective expansions in orders of magnitude, and T, T-l, 7’ and F’, respectively, by

T \

T-‘1 7’

- I f -

iAS - $h”S2 7

(i/6)XySy + . . .,

=

ix2s

. . . )

I

f

-

3x49

+

I-’

(5b)

where I stands for the identity operator and i for v’- 1, and if we equate the coefficients of like powers of X, we obtain the well-known relations (6-8) Ho = Ho, H1’ = HI + i(S.Ho), H; = Hz + i(S.Hl)

- i(S. (S.Ho)),

Ha’ = Ha + i(S.Hs)

- *(S. (S.HI))

-

(i/6)(.5.

(S. (S.Ho)))

H: = H, + i(S.Hs)

- ;(S. (S.Hz))

-

(i/6)@.

(S. (S.Hd)) + 1/24(.S. (S. (5.. (S.H,))))

(6)

UNITARY

for the first transformation,

TRANSFORMATIOS

477

OPERATORS

and

Hot = HI,', HI+ = HI', H,? = Hz’ + i(S.H,‘),

(i,

ZZx+= HI’ + ~(S.HI’), H,,+ = HI’ + i(S.Hz’)-i(S.(S.Hu’)) for the second

transformation

pal vibrational functions

quantum

zl, in the representation

If such resonances

H,’ the off-diagonal

occur,

that

this is possible

then one must

of HI which

operators

for the commu-

such that HI’ is diagonal

must be chosen

numbers

J/O of Ii,). It is well known

occurs.

.S.H,, stands

etc. In these relations,

tator SH,, - H,J’. The S function (or operator)

modify

in all the princi-

of the zeroth-order only if no accidental

the S function

are responsible

waveresonance

so as to leave in

for the resonance(s).

Similar

conclusions appl!, to the S function and Hz +. At the end of the process, the energy matrix associated with the twice-transformed Hamiltonian will include the diagonal matri\elements of Ho, HI’, Hz+, Hat, and H,t and the possible off-diagonal matrix elements responsible

for the resonance(s).

This matrix

is then diagonalized

by numerical

methods.

At first sight this method seems to be very simple. I_Tnfortunately the calculation “b!. hand” of the second members of Eqs. (6) and (7) is estremel!, difficult and leads to ver>. complicated dental

formulas.

resonances

be done things,

These

which

separately

for each

an! extension

formulas

depend

are to be considered coupling

to higher

orders

upon

the type

and the number

of acci-

and a large part of the calculations

scheme.

This

complexity

[we have mentioned

forbids,

in Ref.

has to

among

(I) that

other

such exten-

sions may be necessary,].

The tnainl!,

new formalism characterized

(1) I>eveloped larl!. well suited

we have

developed

by the following with

constant

to high

(3a) or (3b) may be performed (2) Any coupling scheme vibrational resonances high orders. (3) All the operators

reference

speed

(2, 3) to overcome

to programming

computers;

the above

difficulties

consequently,

applications,

it is particu-

any transformation

algebraicall!, (literally) and verl- rapidl!.. ma!~ be considered, including even several

(or rotational,

or both,

involved

in the theory

operator, etc.) are expressed tensors of definite symmetr\..

as linear

In order to keep in view these principal Lion in agreement with the one introduced

is

:

properties

in case of the complete (Hamiltonian, combinations

unitary

of type accidental

Hamiltonian’)

of

transformation

of Hermitian

characteristics, we have adopted in Refs. (Z) and (3).’

irreducible

a new nota-

’ One must rememl~er that we have restricted ourselves to the vibration Hamiltonian throughout this paper. ’ The differences between the notation used in I
478

CHEDIN

AND CIHLA

Let us now recall, with out notation, the well-known formation of an operator, for example, the Hamiltonian HT = exp{iS}H

exp{-

formula

for the unitary

is},

trans-

(8)

where H is the Hamiltonian, S a Hermitian operator and HT the transformed operator which we shall call the transformed Hamiltonian. It is well known (9, p. 287) that Eq. (8) may be written as HT = H + i(S.H)

+ [(Q/2!](S.

(S.H)) + . . . +[(i)“/n!](S.

. . . (S.H)

. . .) + . . . . (9)

This important identity is currently used in quantum mechanics [for example, in relativistic quantum mechanics, the “Foldy-Wouthuysen” transformation (9, p. 814)]. The untransformed and transfomred Hamiltonians may be expanded in orders of magnitude, and we shall write H=

Ho+

HT=

H,$+

HI+ HIT+

-.. + H,+

--a,

. . . + HZ+

(10) . . ..

(11)

On the other hand, H and HT may be expressed in terms of the same set of tensors; the corresponding linear combinations then differ only in their coefficients. We can write

where the set { (rf)jE}, j = 1, . . . N, is the smallest set of distinct irreducible tensors (Hermitian and totally symmetric) in which one can expand both H and HT up to the desired order of approximation. The index rj takes the values 0, 1, 2, . . . according as cr;)jE is a tensor appearing, respectively, in the expansions of Ho, HI, Hz, . . . (or HO*, HIT, HZT, . . .). We shall refer to any of the cri)jE’s as an element of the Hamiltonian basis: { cri)jE), j = 1, . . ., N. The sets {jcTjlH}, {jcTjjHT), j = 1, . . ., AT, are the sets of coefficients appearing, respectively, in the expansions of H and HT over the Hamiltonian basis. We shall refer to any of the icrjlH’s (or ~‘c,,,H~‘s) as a component of the corresponding Hamiltonian (untransformed or transformed) over the Hamiltonian basis. Any of the H,‘s, n = 0, 1, 2, . . ., may then be written in the form

where the symbol C* means that the summation is subjected to the condition rj = n. In the following, and for the sake of simplicity, we shall rewrite the preceding expression as H,, = C j*H j,E, (IS) &

UiYITXRY

where

the running

TRANSI’ORMr\TION

index j, takes the values

tonian

basis over which

tonian

basis

H, is expanded.

will be called:

which

identify

This particular

the H,,-basis

479

OPERATORS

those elements

of the Hamil-

set of elements

(or also, the H,,T-basis).

of the Hamil-

In the

same

wa>. we

shall write H,T = C jnHT j.E. jn Let us suppose

now that

the operator

S of Eq.

(16)

(8) may be expanded

2The reason why the index of the first term of this expansion

is taken

as3

equal to unity

will be clarified

later on.

s= The operator

sz+

s1 +

S may also be expressed

s3 +

(17)

as a linear combination

of tensors.

. ., .V’ the set of tensors occurring in the expansion b!- { (QJ~S}, i = 1, desired order of approximation and by {i(l,jS}, i = 1, . ., A” the these

tensors,

If we denote (17) up to the coefficients

of

we have N’ s = 2 i(&S L=,

(Q’;S.

LVe shall call the set { (rl);S),i

= 1, .

_V’, is the set of components

of S over the S-basis.

3, . according respectiveh-. HJ- anal&

., :I:’ “the S-basis”.

as (Q)~S is a tensor

with

Eq.

(18)

involved

The set ( it7L)S}, i = 1, . . ,’

The index

T; takes

an>. of the SP’s, p = 1, 2, 3, . . . may

(lj),

the values

1, 2,

of S1, Sx, Sz,

in the expansions

.,

be expressed

in

the form

where the running over which

index i, takes the values

S, is espanded.

I;rom Eqs.

This particular

which identif!-

those elements

set of elements

will be called the “S,j-basis.”

of the S-basis

(8) and (17) we have4 by esp (ir.

’ Ie‘orthe sake of simplicity, esp {i(...

H’/‘=

+ Sa+

Ss+

+ SI + Sr + S,) ) hbe nlean

erp ( iS:r)

&}Hesp

Sa+

= esp{i(...

+ Saf

= e~p (i(...

+ &)}H”esp{-

= esp {i(...))H”‘esp

(-

i(S?+

S,)}H’esp{-

{-

i(S1+

i(S,+

S,+

S:j+

esp { iSI )

...)}

...))

...I)

(X1,!

i(..-)}

, where H’ = esp (iS1)H

esp (-

is,},

(21)

an d H” = exp (iS2}H’ esp { -

if&}. . etc.

The transformed Hamiltonian HT ma!. be espanded ing to Eq. (9). If we onl\- consider the first two terms identical

with

H”) :

(22)

in terms of commutators accordof Eq. (17) we have (here, HT is

_

HT = esp (is,)

esp { &}H

esp (-

iS1) esp (-

is,},

(23)

480

CHEDIN

AND TABLE

CIHLA I

and then

HT = H + ,i(&.H) + i&H)

+ [(i)“/2](S1.(S,.H))

+ [(i)‘/2](Sz.

(S1.H)) + [(i)3/6](S,.

(SI. (&.H))) + . . .

(24)

The fundamental purpose of the unitary transformation of the Hamiltonian is to substitute for the matrix representation” of the untransformed Hamiltonian, i.e., an infinite matrix, the matrix representation of the transformed Hamiltonian which consists of a matrix factorized into subblocks of finite dimensions located along its principal diagonal. The dimensions of these subblocks depend upon the coupling scheme. This procedure is applicable to any order of approximation and for any number of accidental resonances (including high-order resonances which are very important for an accurate computation).

5The representation

of the zeroth

order wavefunctions

$0 of Ho (see Eq. (2)).

I I

II

CHEDIN

482

AND

CIHLA

A careful examination of requirements shows that the operator S must satisfy the following rule : The commutator d(1) (S,.H,) must be expressible by a linear combination of the elements of the H,-basis with q = n + p. This can be written as 2/( -

I) ( ipS ’ j,E)

=

C ipj,$JLk~ ,c,E

(23

with q = n + p, where the index h, takes the values which identify the elements of the H,-basis and where the i,jn’$Vkq’sare numerical coefficients which only depend upon the structures of the S-basis and of the Hamiltonian basis. We have shown in previous work (2, 3) how they can be obtained. The set of all possible values of i,, j,, K, and i,j,‘$& # 0 will be called the multiplication table of the elements of the S,-basis by the elements of the H,-basis when the product law is defined by commutators. If we consider the usual case, that is to say, if we suppose that the HI-basis and the Hp-basis, are, respectively, of degrees 3 and 4 with respect to the normal coordinates (the q,,‘s) and to their conjugate momenta (the psC’s) it is found that the &-basis and the &-basis are, respectively, of degrees 3 and 4 with respect to the same operators.‘j Moreover, S must be odd with respect to the p8,‘s whereas the Hamiltonian must be even so as to be invariant with respect to time inversion. We give, respectively, in Tables I and II the most general &-basis and the most general &-basis for a linear triatomic molecule. The components of S over the S-basis are chosen in such a way that (1) After the several accidental (2) After the several accidental (3) ... etc.

transformation (21), H: is diagonal7 (or partially diagonal if one or resonances of the first order are to be considered); transformation (22), Hz” is diagonal’ (or partially diagonal if one or resonances of the second order are to be considered) ;

The presentation of the method for the determination of the components of S over the S-basis is the purpose of the next section. Before concluding let us remark that, starting from Eq. (21) and replacing S1 by hS, H and H’ by their expansions* [see Eq. (lo)] and then H, and H,‘, respectively, b! XnH, and X”H,’ and finally using the identity (9), we obtain the relations (6) by simply equating the “coefficients” of like powers of X. Similarly, we should obtain the relations (7) by starting from Eqs. (22) and (9), replacing Sz by X’%and H’ and H”, respectivel) by H’ and Ht. IV. GENERAL

PRINCIPLE

OF THE METHOD

Let us now consider the second of the Eqs. (6) ; using our notations H1’ = HI + d(-

and conventions,

~)(SI.HO).

6 This conclusion has to be modified if one considers a more complicated H, (or H&basis (for example, when the HI-basis involves tensors of degrees 2 and 3 with respect to the psr’s and the pan’s). The method we present here may be applied without any change. 7 In the representation of the zeroth order wavefunctions of Ho. This requirement explains the form of the expansion of S (see Eq. (17) and note 3). s Limited to H, and H,‘.

From Eqs.

(161 and (25), the Izl-th component

(lo),

of HI’, over the HI-basis,

is given

b>. “l(Hl’)

= “l(H,)

(2;)

x ‘IS jNH ~,;,$W. il ,it

+

If we denote, respectively, b!- HI’ and HI the two column vectors of the components of HI’ and HI over the HI-basis and b\, S1 the column vector of the components of SI over the Sl-basis,

we can put Eq.

(27).into

matrix

form,

as (28)

111’ = HI + MSI, where

the matrix

M of dimensions

EZ,’ and HI is defined

vectors

(:\‘1,:17’) with _I71the common

dimensions

of the two

b!. imj

MB,, il = C ‘JH
that

we know

of HI into that

the matrix

which

transforms

the column

vector

of the

of HI’ :

of the components

H1’ = RHI; then,

we can write

Eq.

Z)HI = MSI,

(R where

(30)

(28) as

I is the identity

matrix

of dimensions

(31)

(,\71,_V1). At this stage

we must

consider

two different cases: Either .V’ = -YI, or _I” < AT1 (the case -1” > _V1 is not possible). the jirst case, the unknown vector S1 is immediatelygiven b> S1 = M-‘(R In the second

case, this vector

-

tn

I)Hl.

(32 )

is ver!. simply, obtained,

b! a least-squares

procedure

&S”

!“I’his is easier result.

than

searching

a system 5’1 =

where

AT is the transpose

of S’ independent

(d&bf-‘~I~(R

-

linear

equations

and leads to the samr

I)Hl,

(A?)

of M.

So the problem of finding the correct S operator, associated with an unitar!. transformation of a given Hamiltonian, becomes trivial as soon as one knows, first, the multiplication

table

of the elements

of the S-basis

by the elements

of the Ho-basis

(when

product law is defined by a commutator) and, second, the matrix which transforms vector of the components of HI into that of HI’ I”. The iirst of these two problems been discussed

in detail

in Ref.

(3). The second

problem

is discussed

the the has

in the next part

of

this paper. V. DETERMINATIOS

I;rom now on we limit molecule. The method

for finding

form the HI-basis

expressed

our stud!. the matrix

01: THE

to the vibration

M.YTRIX

Hamiltonian

K may be summarized

in terms of the normal

R

as follows:

coordinates

1”.Ind the matrix which transforms the vector of the components iml)lies that we limit the expansion (17) IO the first two terms).

of

of a linear

triatomic

We first trans-

~1, 421, q2?and q3and their

Hz'into that of Hz’ (this notation

4x4

CHEDIX

AND CIHLA

conjugate

momenta

corresponding

p,, ~21, pz2 and pa to an equivalent

raising

and lowering

.W = 1, 21, 22, 3, there

operators.

corresponds

HI-basis

Let us recall

the raising

expressed

that

in terms of the

to each pair

(y,,,p,,,,

operatol

ll,, = (v2)P’(ySC + i&J/z), of which the onl!. nonvanishing

matrix

elements

(31)

are of type”

(z,ll~~ +

l), and the lower-

ing operator a,<,+ = (\‘2)-‘(q,, of which the onI!- nonvanishing

matrix

-

elements

ip,,/lz),

(35)

are of type”

(?-1.~!11~ - 1). The commuta-

t ion laws of yA,, and p,,, and of aso and aaot are, respectively, yw(p.%Jtil as&s, \I’e shall denote

the matris

espressing

the elements of the HI-basis are taken as row vectors:

(~,,,u,,~)

(3b.a)

(psJh)qgc = + i, t -

as0t usr = +

the elements

(36.h)

1.

(qs6,ps,,)in terms of

of the HI-basis

by /,I. One must

notice

that

the two HI-bases

E,, = C ,,F ~li,(W, II where

an element

of the HI-basis

It is immediately

seen

that

(u,s,,a,,‘) in terms of the elements dimensions of these two matrices in Table

IlI,

this Table,

the structure

p,, stands

Knowing

the matris

new HI-basis The matrix

is denoted

matrix

by ,,F.

expressing

the

elements

(qss,p,,) and of the HI-basis

and the numbers

In

respec-

a~+ and a;$+.The two HI-bases involve oni\- Hermitian in the C‘,, point group and symmetric with respect to time

awl+,

L1, we can immediately

(ti,s,,u,Y,t) from its espression

hand,

HI-basis

(a,,,~,,+).

1, 2, 3, 4, 5, 6, 7 and 8 stand,

derive

the expression

over the HI-basis

the espression

for H, over

the

(qS,,,pSb).

expression of HI over the HI-basis (ti,so,asat) then brings elements (all off-diagonal in the v,<‘s) of this operator.

On the other

of the

of the HI-basis (yS,,,pSC) is the inverse LIP1 of L1. The are equal to the dimension of the HI-basis. U’e give,

of the HI-basis

for p,,/Jz

tively, for al, ~22~,a22, u3, al+, operators totall! symmetric inversion.

(ase,~,,t) the

of HI’ over the same basis

into must

operator(s) which is (are) responsible for the accidental resonance(s). the case of the carbon dioside molecule, and since only the strong

evidence

all the

onlj. involve

the

For example, in Fermi resonance

occurs (WI- 2w?), the expression of HI’ over the HI-basis (~~,~,,cl,,,+)ma\- onI\. contain the element number 1X of this basis, i.e., according to Table III, _ _ 166 + 22.5 + the matrix

elements

of which

177 + 335,

are of the type

(37)

486

CHEDIN

AND

TABLE

CIHLA IV

In the case of nitrous oxide one would find only the elements responding, respectively, to the two Fermi resonances

number

18 and 12 cor-

and

The latter case was considered in order to test the method. Under these conditions, one has to go from only the operators involved in the expression of H: over the H&ash (asa,ass+) to their espressions in terms of the operators involved in the original HI-basis ((I~~#~~) so as to get the expression of H1’ over the same basis. This operation is performed by the matrix (Ll?)* where the asterisk indicates that, in the matrix LIP’, all the columns corresponding to the operators which must not appear in the expression of HI’ over the HI-basis (aso,asaf), have been set equal to zero. One will immediately verify that the matrix R, defined in part IV of this paper, is given by the product R = (L-‘)*(L).

(38)

All of the above considerations may be applied to the transformation Hi + H2’.13 The only modification originates from the fact that the expression of H2T over the Hz-basis (a,,,a,,+j may involve the operators diagonal in all the vs’s in addition to the operators responsible for the accidental resonances (in the case of NzO, for example, one has to consider a second order accidental resonance ; see below). VI. DETERMINATION

OF THE

MATRIX

L,

We refer the interested reader to Appendix B of Ref. (3). He will find there the details about this particular step of the problem we are considering here. VII.

RESULTS

We give, respectively, in Tables IV and V, the numerical values of the components of S1 and St over the S-basis (see Tables I and 11) for the vibration Hamiltonian of the 13This notation

implies

that we limit the expansion

(17) to the first two terms.

UNITARY TRANSFORMATION

OPERATORS

487

TABLE V

i,

-

i

1

2

2

0.u

?C

2S

33

5.563

34

0.0

35

-2.504

1O-3 10-3

3"

7.511

3.!lo4

10-3

37

0.0

6.527

?O-3

38

0.0

39

4.864

0.0 0.0 ->.397

10-J

?/

0.0

41

-2.734

1O-4

42

6.546

10-3

43

0.0

1."

44

-2.256

2s

5.721

1O-3

45

0.0

3c

4.295

10-3

.;6

0.0

31

0.0

-17

4.111

10-4

10-3

40

26

3: -

1O-4

48 --

lO-3

-9.189

10-4

1.313

10-3

carbon dioxide molecule (for the species W602). The values of the molecular constants we have used are given in Ref. (1). This computation takes into accout the strong Fermi resonance of the first order [see Eq. (37)]. Our results are identical to those which can be derived from the formulas Ref. (II)” for a single, first-order, Fermi resonance.

given in

VIII. COMPUTATION TIME

The matrix R is computed once and for all for a given wwleculd5in less than 5 set with the 6600 C.D.C. computer. This is the maximum time required in the case of the nitrous oxide molecule second-order

using

a second contact

transformation

which takes into account

the

resonance

The matrix elements of R are stored on magnetic tape. The computation of the components of S never requires more than a few seconds with the same computer. These very short times demonstrate, more than anything else, the great effectiveness of the method. RECEIVED: February

25, 1972

I*Part VII of this paper is in error. It has been recently corrected by G. Amat (see Ref. [lZ]). 1sThis matrix is the same for all the isotopic species of the molecule.

488

CHEDIN

AND

CIHLA

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