Derivation of the nonrigid rotation-large-amplitude internal motion Hamiltonian of the general...

JOURNAL OF MOLECULAR SPECTROSCOPY l&24-6 1 (1988)

Derivation of the Nonrigid Rotation-Large-Amplitude Internal Motion Hamiltonian of the General Molecule

VIKTOR SZALAY*

Research Laboratory for Crystal Physics, Hungarian Academy of Sciences, P.O. Box 132, H-1502 Budapest, Hungary: and Herzberg Institute OfAstrophysics, National Research

Council of Canada, Ottawa, Ontario, Canada KIA OR6

The nonrigid (effective) rotation-large-amplitude internal motion Hamiltonian (NRLH) of the general molecule with one or more large-amplitude vibrations has been derived to the order of magnitude x2Tvar. The derivation takes advantage of the idea of a nonrigid reference configuration and uses the contact transformation method as a mathematical tool. The NRLH has a form fairly similar to that of the effective rotation Hamiltonian of semirigid (i.e., normal) molecules. From a careful examination of the Eckart-Sayvetz conditions and of the Taylor expansions of the potential energy surface in terms of curvilinear displacement coordinates, three types of large- amplitude internal coordinates of different physical meaning (effective large-amplitude internal coordinates, real large-amplitude internal coordinates, and reaction path coordinates) are described. To test the ideas and the formulas the effective bending potential function of the C3 molecule in its ground electronic and ground stretching vibrational state is calculated from the ab initio potential energy surface given by W. P. Kraemer, P. R. Bunker, and M. Yoshimine (J. Mol. Spectrosc. 107, 19 I-207 (I 984)). The calculations were carried out by using either the effective or the real large-amplitude bending coordinate of C,. The NRLH theory is compared to the nonrigid bender theory at a theoretical level as well as through the results of the test calculations.

I. MOTIVATION

The complete analysis of the spectra of a number of molecules, for example, Cs, C302, HCNO, NH3 , CH2, and (CH&O, needs a nonrigid rotation-large-amplitude internal motion Hamiltonian (NRLH) but only the simplest cases have been treated up to now. General expressions have been given by Sorensen (I), but the degenerate vibrations were excluded in his derivation. General expressions relating the large- amplitude internal coordinates of the real molecule to the large-amplitude internal coordinates and the dimensionless normal coordinates of the nonrigid reference molecule, j = j(p, q), are unknown; thus separate computer programs have to be written for almost all molecules.

The extended Hougen-Bunker-Johns theory (referred to as EHBJ) as introduced in Ref. (2) includes all of the theories based on the minimization of the coupling of rotation large-amplitude internal motion momenta. Thus it can be applied to almost all nonrigid molecules. In fact, it has been shown (2) that the EHBJ theory includes as special cases the formulas of the original HBJ theory, the internal axis method (IAM) of internal rotations, and Malloy’s method on the puckering of small ring

* NRCC Research Associate.

24

0022-2852188 $3.00

THE NONRIGID ROTATION-LAM HAMILTONlAN 25

molecules. Recently it has been shown (3) that Pickett’s Hamiltonian (4) (the corresponding theory is frequently referred to as the reduced axes method) is again a special case of the EHBJ Hamiltonian. However, the EHBJ theory not only revises the earlier theories but also helps to answer problems which were too difficult to solve by the earlier theories. For example, the IAM Hamiltonian of an internal rotor consisting of completely asymmetric top and frame can be easily derived (5). To my knowledge this last problem has not been solved yet, although some attempts have been made (6).

Such a general theory as that leading to the EHBJ Hamiltonian should be developed as precisely as possible. This paper is organized and written to give a clear discussion of the new physical ideas of such a development sacrificing lengthy but straightforward mathematical details and detailed discussions of some important but known physical ideas. Coordinates for describing the dynamics of nonrigid molecules are introduced and a Taylor series approximation to the exact form of the rotation-large-amplitude internal motion vibration Hamiltonian is given in Section II. Following an appropriate small-amplitude vibrational contact transformation the formulas of the NRLH of the general molecule are derived in Section III by a small-amplitude vibrational averaging procedure. To suit this operator to ab initio calculations of rotation-large-amplitude internal motion spectra an original method is suggested for the calculation of isotopically dependent force constants from known isotopically independent ones in Section IV. This method is then compared with the nonrigid bender theory (7, 8) (Section V). Following a discussion on the geometry relaxation (Section VI) a summary and discussion of the theoretical results of Sections II-VI is given (Section VII). The NRLH theory is tested and developed through calculations on the C3 molecule (Section VIII). Section IX summarizes.

11. EXPANSION OF THE EXACT ROTATION-LARGE-AMPLITUDE INTERNAL MOTION VIBRATION HAMILTONIAN ACCORDING TO THE DIMENSIONLESS NORMAL

COORDINATES OF SMALL-AMPLITUDE VIBRATIONS

Within the Born-Oppenheimer approximation the shape and internal dynamics of free molecules are entirely determined by their Born-Oppenheimer potential energy surfaces. The most natural set of coordinates describing a Born-Oppenheimer potential energy surface of all but extremely flexible molecules consists of molecular geometry parameters such as bondlengths, bond angles, dihedral angles, and so on, i.e., the traditional parameters used by chemists. Molecular spectroscopy experiments provide a great deal of information on the potential energy surfaces of molecules, but this information cannot be easily explored and analyzed. Already at the first stage of the theoretical part of that work we get into trouble. What functional form

v= V(/(is,,&,. . . ,&N-6)=? (1)

should be chosen for the potential energy function of an N atom molecule expressed in a set of natural curvilinear internal coordinates j, , & , . . . ~ &N_~?

Functions for individual stretching (9) and bending (10, II) motions have been suggested, but there is no hope for finding a general function including all curvilinear internal coordinates as variables. For semirigid (i.e., normal) molecules, instead of searching for such a function an assumption and an approximation is made. It is

26 VIKTOR SZALAY

assumed that internal motions can be described by small-amplitude curvilinear displacements II,, X2, . . . , A~,+G, which are measured from a reference configuration defined by a set j?, jq, . . . , j&+6 of curvilinear internal coordinates

H3N-6 = ?3N-6 - i :N-6, 1A3N-61 e k$N-61. (2)

The exact potential energy surface then can be approximated by its Taylor series expansion truncated at a given term, say, at the fourth term

(i,j,k,Z= 1,2,. . .,3N-6). (3)

A Hamiltonian for the nuclear motions can now be set up in terms of these curvilinear displacement coordinates, by deriving the corresponding formula for the kinetic energy operator. However, the latter is difficult to derive and the Hamiltonian will include a very complicated kinetic energy operator.

To achieve a simplification in the kinetic energy expression a new set of displacement coordinates, the set of linearized internal coordinates, is introduced. The dimensionless normal coordinates ql, q2, . . . , q3,+6 are a special type of such coordinates. Their relation to the curvilinear internal displacements has been established (12) and is given by a nonlinear transformation

A3N-6 = CLjN-6qr+; c’%N-6@qs +; 2 L%6@@h + $ 2 L%6qr~d?t~u I rs rst i-stu

(r,s,t,u=1,2 ,..., 3N-6) (4)

or in the compact notation of Ref. (12)

II = L*q. (5)

The matrices L given in Eqs. (4) and (5) are slightly different from those of Ref. (12),

THE NONRIGID ROTATION-LAM HAMILTONIAN 27

as we use dimensionless normal coordinates instead of dimensioned ones. They are related by the equation

(6)

where O, denotes the frequency (in cm-‘) of the rth normal vibration. The Hamiltonian obtained through the coordinate transformation of Eq. (5) has

turned out to be particularly useful in the analysis of the rotation-vibration spectra of a great many molecules. Whenever difficulties arise in applying this exact Hamil- tonian one can be sure that some of the basic assumptions referring to the smallness of internal displacements in Eq. (2) are broken. In the case of molecules having large- amplitude internal motions some inequalities in Eq. (2) are broken and the expansions in Eq. (4) and the expansions of the potential and kinetic energy according to curvilinear displacements or normal coordinates fail to converge or converge very slowly. Thus a new Hamiltonian must be developed for molecules with large-amplitude internal motions.

It is clear that the large-amplitude coordinates pi, j2, . . . , jSLcc3N_6j, say, have to be treated explicitly and normal coordinates can be introduced for the small-amplitude vibrations. Such a description still employs the concept of a reference configuration. However, we have a nonrigid reference configuration defined by the coordinates pl,

P2, ...3 PL, is+,, PoL+2, *. ., j&b, where the internal coordinates pi , p2, . . . , PL

corresponding to certain large-amplitude coordinates ofthe real molecule are not fixed at a given value but are considered variables. When the small-amplitude displacements are equal to zero it is required that

PI = PI P2 = P2

PL = PL.

Instead of the instantaneous curvilinear internal coordinates, pi, pz, . . . , ,G3N_6, 3N - 6 - L linearized displacement coordinates and L effective large-amplitude coordinates enter into the kinetic energy expression automatically if the standard methods for deriving a rotation-vibration Hamiltonian are applied. It is not self-evident but follows from earlier works (8, 13-15) that the exact form of the rotation-large-amplitude internal motion vibration kinetic energy operator is

s T= /A”~(J, - IIa)polSp-“2(Jg - II,&1’4 + l”4 2 yt’2pr~-“Zy~‘2p,~“4. (8) I

Here and throughout the paper the Einstein summation convention is used whenever a Greek letter occurs twice in a given term, but the summation symbols are retained for sums over atoms and normal coordinates. T is in centimeters-’ J, and II, are in units of A, and p, is the momentum operator conjugated to the dimensionless normal coordinate qr. The Greek indices CY, /3 run over x, y, z, p,, ~2, . . . , and pL, where x, y, z denote the axes of the molecule fixed coordinate system MS0 (see Ref. (2)). The system MS0 and the effective large-amplitude coordinates are unambiguously deter-

28 VIKTOR SZALAY

mined by the instantaneous position of the molecule in the laboratory system and by the Eckart-Sayvetz conditions

7-= CmmdfSo=O m

%= Cmma~SoXd~So=O m

(9)

P, = Cm,(aa~so/apl).d~so=O M

PL = 2 m,(da~So/dpL)~ dESo = 0, m

(10)

where dgSo means the Cartesian displacement vector of the mth atom of the molecule and it is expressed in terms of coordinates derived in the system MS’. Since the results obtained are equally valid whether or not the momentum couplings between the rotation and the large-amplitude internal motions are minimized, no conditions, in addition to the Eckart-Sayvetz conditions, are introduced.

In contrast to the case of the kinetic energy it is the instantaneous curvilinear internal coordinates, pi, j2, . . . , j3N_6, and not the linearized displacements and effective large-amplitude internal coordinates that enter into the potential energy expression in a natural way. Thus the relations connecting the coordinates pl , p2, . . . , pL, ql,

q2, . . . 9 q3N_6-L to the instantaneous curvilinear internal coordinates must be determined so that the potential energy surface can be described in terms of the coordinates that the kinetic energy operator has been written in. We put forward the results proved in Section IV. The exact potential energy surface of a molecule with L large-amplitude internal motions can be approximated as

vh,i52,. . . ,ij3N-6)

where

pL+l =i5t+,,i’L+,=i%+2,. . . ,?3N-6=&+6). (12)

r, s, t, and u have values 1, 2, . . . , 3N - 6 - L, V is in centimeters-‘, and the force constants %, w,, %, and a,,,, are functions of the effective large-amplitude internal coordinates.

A useful approximation to the kinetic energy operator can also be obtained by rewriting Eq. (8) in a more convenient form,


(where P = PI, PZ, . . . , pL and the operators enclosed by square brackets act only within these brackets), and expanding it according to the dimensionless normal coordinates following the method of Ref. (15).

TABLE Ia

Terms in the Rotation-Large-Amplitude Internal Motion Vibration Hamiltonian Arranged by Order of Magnitude and Powers of J

a

KOT”IB K’T VIB

KZT VIB

K3T VIB

-[ JpB;,& 1

J’ -2B’ II J lx6 a 6

I

Note. pr, q,, and J,, n, are dimensionless operators. LI, p = x, y, z, p. p = pI, p2, . . , p,. r, s, t, u = I. 2, .3N - 6 - L.

30 VIKTOR SZALAY

TABLE lb

Definition of the Rotation-Large-Amplitude Internal Motion Coefficients Introduced in Table Ia and the Detailed Expression of the Pseudo Potential Term, Cl,,

+ i 1 LTr(h~)Tr(h~)-2Tr(hfh~)]} r

Then by the use of the ordering scheme of Ref. (7) the terms of the expanded Hamiltonian can be arranged as it is given in Table I. All the terms of the order of magnitude ~~~~~~~ K’T~,~, and KIT VIB, TVIB denoting a typical small amplitude vibrational energy, are included in Table I. The KIT VIB terms noted are necessary if one wants to use Hermitian centrifugal distortion and Coriolis operators. All the other terms of the order of magnitude KIT VIB or higher are neglected. The derivation of the formula of the pseudo potential term U. is given in the Appendix.

III. DERIVATION OF THE NONRIGID (EFFECTIVE) ROTATION-LARGE-AMPLITUDE INTERNAL MOTION HAMILTONIAN OF THE GENERAL MOLECULE

The NRLH of the general molecule is developed up to the order of K2 in this section. Using perturbation theory the exact rotation-large-amplitude internal motion vi-

bration energy, eRLV, in a given electronic state can be written as

= Ek_v + (RLVIH,IRLV) + c’ “R;;;t”l’;Ev;;“2 + (RLVIH21RLV), (14) R’L’V’

where EgLV = (RLVjHojRLV). The operators Hi are the sums of operators having


order of magnitude K’TVIB given in Table I. R, L, and V include the quantum numbers describing the eigenstates of the zero-order Hamiltonian, HO, and

HOIRLV) = (HtL + H,v)IRL)I I’) = (E:L + Eo,)IRL)I V) = E$Lv/RLV). (15)

The term

however, still includes some higher order contributions that can be neglected. Each of these contributions is smaller than, or of the order of magnitude of, K~TVIB and occurs for the operators J,, , Jp2, . . . , JpL which do not generally commute with +‘r, arst, BO,,, B&. In fact, for example,

= B~8[J,B&1qrs,JsJ,,J~(-K3TvIB)+B~~B~yqrqSJpJ~JaJ~(NK2TvIB), (16)

where p f p is assumed. Assuming the inequality

EEL - E;cLt 6 Et - E$ (17)

one can define an effective operator H,rr for which the diagonal matrix elements (RLIH,~lRL) are good approximations to ERLV.

The preceding discussion may immediately lead to the application of the van Vleck transformation (16). However, as far as an effective Hamiltonian with accuracy K*T,,,~ is to be derived

[Jo, %?I= 0

[J,, B&l = 0

[J, > @rsd = 0. (18) As a result of these equations and Eq. (17) the method of contact transformation can also be applied without difficulties. We recognize that the ordering scheme given in Table I is just the Nielsen ordering and the S operators (see Table II) of the first small- amplitude vibrational contact transformation can be immediately determined and applied as usual. Since principles and examples of applications of the contact transformation technique have been described in several books and articles, see, for example, Ref. (I 7) and the references therein, the results of our calculations will be given here without detailed discussions.

Carrying out the first contact transformation one obtains the nonrigid rotation- large-amplitude internal motion Hamiltonian of the general molecuie with accuracy K2TvIB aS

~~,=(VIHIV)=(VIHo+~,+HzlV)+ti(VI[S,,H,+Ij,]lV), (19)

32 VIKTOR SZALAY

TABLE II

S Operators for the First Small-Amplitude Vibrational Contact Transformation

where HI = H, + i[S* ) HoV] (20)

s, =slo+s30+&, +s,2. (21)

The calculation of these averages then yields the expression

&= B$J,J@+ r ,8,aJ~JsJ,Ja+[J,B~~lJ,-2B88~J~+ Vo+ uo+ u,

S, S’ run over nondegenerate vibrations and t, t’ run over doubly degenerate vibrations. The rotation-large-amplitude internal motion vibration functions B$, reSTa, {",

U*,d, * * . ,f; resulting from the contact transformation and the vibrational averaging procedure are related to the potential energy surface, Eq. (1 I), of the molecule and to the geometry of the reference molecule through the equations

(23)

34 VIKTOR SZALAY

X

+ f@‘:bhPb + eh’bt’fb + @P:bPbPb) WtWt’Ot” + & 2 assss + ; n tt’t” s

X & C (3Cztatata + 6% t

tatntbtb + jatbtbtbtb) - ;B:g 2 i-,4 S% - ; B$ S<d

x 2 (i-,4, i-,B,a + S-:tb i-:tb) - ;B:o 2 (!%?a i-ih + &‘b i-k’6 + &‘a !%bt’, + j$,t’b i-&b)

St f<l’

(26)

Cd:=U,-;C@s4s+ c (ata %ss + @tb%bss) & (27) s’ Us’ t

d = wt - ; 2 @‘s(+Pstata + @‘stbtb)’ s WS

- ; 2 (%‘a %atata + %‘a *t’atbtb + +’ t’b @ t’

t’btatn + +t’h@t’blbtb$, (28)

36 VIKTOR SZALAY

where the indices s, s’, s” run over nondegenerate vibrations, 1, t’, t” run over doubly degenerate vibrations, ta, tb denote two components of the tth doubly degenerate vibration and

o,, = (w, + ws + 01)(-w, + w, + wJ(wr - ws + wJ(w + 0s - 4. (37)

Only nondegenerate and doubly degenerate vibrations are introduced in the final formulas, Eqs. (22)-(36), but if a problem needs triply degenerate vibrations they can be introduced as well. Even the Fermi and Coriolis resonances can be built into the effective Hamiltonian. Symmetry relations have not been used in writing the NRLH.

The results are very similar to those obtained for the effective rotation Hamiltonian (NRH) of molecules having no large-amplitude internal motions. This is not surprising if the formal similarities of the Hamiltonians of semirigid and nonrigid molecules as well as the formal similarities of the S operators of the first contact transformations leading to the NRH and NRLH, respectively, are considered.

There is a convenient way of partially checking our formulas for the NRLH by reducing them to the NRH case and comparing the formulas obtained in this way with the formulas of the NRH derived earlier by several authors (18). In another way of checking we applied our formulas to triatomic molecules and compared the results with those of Ref. (8). Almost complete agreement was found between the formulas obtained by contact transformation and the formulas obtained in Ref. (8) by the van Vleck perturbation method. The only difference is that we have no quadratic terms having the order of magnitude K’ T VIB in the potential energy function, Eq. (1 l), and thus the terms given in the eighth row of Eq. (5.29) in Ref. (8) are missing from our effective potential function. It will be shown in the next section how the quadratic terms of the order of magnitude K’ TVIB can be eliminated.

IV. THE POTENTIAL ENERGY

In this section a method is developed for the calculation of the potential energy surface of a nonrigid molecule, Eq. (1 l), given in the normal coordinate representation, i.e., for the calculation of the force constants ar, w,, a,,, arslU, from the potential energy surface of that molecule given in the small curvilinear displacement representation, Eq. (39), i.e., from the isotope substitution independent force constants (eV/ dji)o, (a2v/$&asj)o, (a3~//lajid~$&)o, and (8V,ap$p$pkapl)o.


First, the basic equations for curvilinear internal displacements, Eq. (2), are rewritten for molecules having L large-amplitude internal motions as

AI=;,-PI, l~Il4P,l

~2=i%--P21 lN4P*l

fl3N-6 = PIN-6 - P!N-~ 7 1~3,‘+614;IS!N-61. (38)

The potential energy surface of the nonrigid molecule considered can now be approximated by a truncated Taylor series expansion about the nonrigid reference configuration as

lJkl ’ J k ’ 0

which is formally the same as Eq. (3) obtained for molecules having no large-amplitude internal motions. However, Vo and the force constants (aP’/&5;)0, (a’J’/ai;iai;ilo, (d3v/d&d~j&)0, and (d4T/ld&$jd&d~,)0 are not COnStaIItS here but fUnCtiOIIS of the effective large-amplitude coordinates.

The transformation of dimensionless normal coordinates to the curvilinear displacements can be written as

II=L*q, (40)

again, observing that there are 3N - 6 curvilinear displacement coordinates and only 3N - 6 - L normal coordinates now (see also Ref. (I)). If the small-amplitude curvilinear displacements BL.+~, (IL+z, . . , 113N-6 are equal to zero the normal coordinates should be equal to zero as well and so the restrictions, Eq. (7), are satisfied. Substituting the curvilinear displacements by normal coordinates in Eq. (39) and comparing the result of the substitution with Eq. (11) one can establish the relations

Gr= 2 LV ( 1 i +i o

L: (41)

(42)

(43)

38 VIKTOR SZALAY

(r,s,t,u=1,2 )... ,3N-6-L,i,j,k,Z=1,2 ,..., 3N-6). (44)

Now what should be done is the determination of the elements of the tensor L. To make progress in solving this problem the origin of Eq. (40), and a hidden property of the potential energy function, Eq. (1 l), should be clearly understood. A brief account of these follows.

The curvilinear displacements can be expanded according to unrestricted Cartesian displacement coordinates

where the indices m, n, o, p correspond to the atoms of the molecule and run from 1 to N, and the Greek letters denote the Cartesian components X, y, z. Combining the equation

T;‘12qr= Cm$‘l dt.,@ am,r rrm (r= 1,2,. . . , 3N-6-L;m= 1,2,. . .,N;(~=x,y,z)

m

(46)

with Eqs. (9) and (lo), inverting the complete set, and putting the nongenuine internal coordinates “r, R, ‘PI , . . . , and ‘PL equal to zero, i.e., applying the Eckart-Sayvetz conditions, the relation

d,, = d!$ = 2 m;1’2 Ia,,rr; ‘12qr

(I-= 1,2,. . .) 3N-6-L,m= 1,2,. . . ,N,a=x,y,z) (47)


results. Moreover the substitution of this new expressions of Cartesian displacements into Eqs. (45) leads to formulas for the elements of the tensor I,

LJ = 2 B~mm~1’21rm,r~;‘~2 (48) m

L:” = ? Bpm~~“(mmm,)~‘~21,,,~Z~n,s(~r~s)~1~2 (49)

LT’ = c Bpm~B”~Y0(m,m,m,)~“2f,,.,~~n,s~ro,r(~r~s~~)~”2 (50) mno

LY = C B~m~fin~y0~ap(m~mnmomp)~‘~2~~~~~l~~~~~~,~l~~,~(~~y~~~~~)~1~2 mnop

(r,s,t,u=1,2 )...) 3N-6-L;m,n,o,p= 1,2,. . . ,N,a,@,r,S=x,y,z;

i= 1,2,. . . ,3N-6). (51)

Then turning to the potential function, Eq. (1 I), one can state: In applying a potential function given in Eq. (11) the Eckart-Sayvetz conditions are satisfied and the Cartesian displacements are related to the normal coordinates by Eq. (47).

Now, having discussed the origin of Eq. (40) as well as an important property of the potential function, Eq. (1 l), the calculation of the elements of the tensor L can be completed.

By the use of Eqs. (42), (48), and (49), the normal frequencies may be related to the isotopically independent forces as

(r,s= 1,2,. . . ,3N-6-L;i,j=1,2 ,..., 3N-6;m,n=1,2 ,..., N,ty,fi=x,y,z),

(52) where the quantity enclosed by square brackets can be recognized as the quadratic Cartesian force matrix J Virtually the eigenvalue problem of the mass weighted f matrix, F, provides the normal frequencies of the small-amplitude vibrations as well as the matrix 1 needed to calculate the tensor L. In reality this is not the case. Let us remember that in deriving both the kinetic energy operator, Eq. (8), and the potential energy operator, Eq. (1 I), a new set of orthogonal coordinates were introduced and then the Eckart-Sayvetz conditions were applied. Due to the Eckart-Sayvetz conditions and the orthogonality of the new coordinates the matrix elements I,,,, should satisfy a number of sum rules:

C m$’ lam,r = 0

e,@, 2 ml2 aFZ”lvm,, = 0 m

C m~2@aZ!!o/ad1,m,, = 0 m

C mX2(aa%o/adI,,,r = 0 m

(53)

.=

VIKTOR SZALAY

: [a,, - (m,m,)“2M-1]&B - e,,,e~ar((Z~so)~‘),r(m,m,)‘~2a~~oa6M,So

ij=l

(r,s= 1,2,. . .) 3N-6-L;m,n= 1,2,. . . ,N;a,p,r,6,~,!:=x,y,z), (55)

where ea0, denotes an element of the Levi-Civita tensor, and IFso and M are the 3 + L-dimensional generalized tensor of inertia and the mass of the reference molecule, respectively. It has just been discussed that the special choice of the potential function, and so Eq. (52), implies the relation Eq. (47) transforming the normal coordinates to the Cartesian displacement coordinates. Writing this expression into Eqs. (9) and ( 10) and using the condition that all the nongenuine internal coordinates are equal to zero the relations Eqs. (53) turn out to be satisfied. Equation (54) is naturally satisfied due to the orthogonality properties of the eigenvectors. Thus we can state: If the Hamil- tonian operator of a molecule is determined by the use of the Eckart-Sayvetz conditions, the sum rules given in Eqs. (53) and (54) will be automatically satisfied by the corresponding elements of the eigenvector matrix of the quadratic mass weighted Cartesian force matrix, F, but there is no reason to suppose that they will satisfy the sum rule expressed by Eq. (55).

On the other hand if the matrix elements l,,,r satisfy all the sum rules, Eqs. (53)- (55), they will also satisfy the equation

@J&s = (r,~s)-“~ C [C ~3yp,amFam,~rn~3~n,60]1~~,,I~o,s (56a) op mn

or

where

(56b)

(r,s= 1,2,. . . ,3N-6-L;m,n,o,p=1,2 ,..., N;a,/I,y,6=x,y,z). (56~)

This can be derived from Eq. (52) by using the sum rules given in Eqs. (54) and (55). Reversing the order of ideas we can say that the matrix Z, the elements of which satisfy all the sum rules, can be calculated by diagonalizing the 3N X 3N dimensional matrix 3. As a result of the solution of this eigenvalue problem the squares of the force constants w, are also obtained.

Having determined the matrix 1 the elements of the tensor L can be calculated by making use of Eqs. (48)-(5 1) or the more convenient expressions derived in Ref. (Z2). Since in that paper the calculations were extended only to the third-order elements, LFt, and our theory needs the fourth-order elements, LYt“, as well we should calculate them for the basic types of curvilinear internal coordinates. The elements By” and


fjpm,Bn which are necessary for the calculation of the matrix I can be obtained as described in Refs. (19, 20).

With the development of the formulas of the NRLH of the general molecule and the ab initio quantum chemical methods worked out for the calculation of the Born- Oppenheimer potential energy surfaces of molecules we have acquired all the necessary knowledge to carry out accurate ab initio calculations of rotation-large-amplitude internal motion spectra of molecules having one large-amplitude internal motion.

The NRLH has been formulated for molecules with several large-amplitude internal motions but in these cases some mathematical difficulties may arise in solving the Schrodinger equation of the NRLH. There are no such difficulties in the case of molecules having a single large-amplitude internal motion. Furthermore a general formulation of the generalized tensor of inertia has also been worked out (2) for these molecules.

V. COMPARISON TO THE NONRIGID BENDER HAMILTONIAN

The formulas of the NRLH of the general molecule have been applied to triatomics. It is found that the results are almost in full agreement with those of the nonrigid bender (NRB) method (8). The reason for the only difference can be immediately understood by consulting Eq. (52).

Sorensen proposed an alternative scheme for assigning orders of magnitude to terms of the expanded Hamiltonian (l), and we can compare his formulas to the formulas given in Section III as far as molecules having no degenerate vibrations are considered: there are no differences.

There was no formulation of the NRLH including doubly degenerate vibrations prior to our work. The present formulation could be achieved relatively easily by applying the method of contact transformation. The authors of earlier works (1, 7, 8) applied the van Vleck perturbation method. This is merely a technical difference. An essential difference in physical ideas, however, can be seen as the derivations of the potential energy function, Eq. (1 l), are compared. Sorensen’s derivation is partly based on the more comprehensive treatment of the NRB theory (8). In the latter method the potential energy surface, instead of being expanded around a nonrigid reference configuration as in Eq. (39) is expanded around an equilibrium configuration.

It is the definition of the effective large-amplitude internal coordinates pi (i = 1, 2, . . . ) L) which has played a central conceptual role in establishing both the NRB (8) and the NRLH theory. According to Ref. (8) the coordinates pi are thought as mass- dependent coordinates defined by the Eckart-Sayvetz conditions. The NRLH theory takes advantage of another view of defining effective large-amplitude internal coordinates. The effective large-amplitude internal coordinates are thought (see Section II) as geometrically defined internal coordinates of the nonrigid reference molecule, which satisfy Eq. (7) at zero small-amplitude vibrational displacements. This is the phenomenological definition of the large-amplitude internal coordinates. According to the NRLH theory the mass dependence is introduced into the expansion of the potential energy surface through the Eckart-Sayvetz conditions leading to mass-dependent displacement coordinates 5Ii (i = 1, . . . , L) and d?$. That is, the Eckart- Sayvetz conditions give the dynamical definition of the large-amplitude internal co-

42 VIKTOR SZALAY

ordinates (for further details see Appendix B). By the use of the Eckart-Sayvetz conditions, relations of the type

(i= 1,2,. . . ,L,j,k,l,m=L+l,L+2 ,..., 3N-6) (57)

can be derived for any molecule, at least in principle, from which approximate expressions of the Iii displacements can be obtained:

(i= 1,2,. . . ,L;j,k,l,m=L+ l,L+2,. . . ,3N-6). (58)

The A coefficients are functions of the internal coordinates and the masses of the atoms of the reference molecule.

There are three ways for introducing effective large-amplitude internal coordinates to describe potential energy surfaces. One has been given in Section IV and Eqs. (57) and (58) provide two others: (a) Let us expand the potential energy surface as

+ $ ,c Fjkh~jflk%% (j,k,I,m=L+l,L+2 ,..., 3N-6) (59) Jkh

and substitute ji (i = 1, 2, . . . , L) by their approximate expressions Eq. (57). This can be done in some special cases (see, for example, Ref. (8)), but not in general. The following method is applicable in any case. (b) Let us substitute the approximate expressions of Iii - s Eq. (58) into Eq. (39). No matter which procedure, (a) or(b), is applied the result is the approximate formula

Vr Vo(P1, P2,. . . 9 Pd + c SjAj f i g gjkkJilAk+ d c sjk/gjflk%

j Jk Ikl

1 + ~j~m~jklm~j~k%%n (j,k,Z,m=L+l,L+2 ,..., 3N-6), (60)

which has always been cited in papers dealing with large-amplitude internal motions of molecules. The coefficients of this expansion are very involved functions of the isotopically independent force constants, the masses of the atoms of the molecule, and the geometry of the nonrigid reference configuration. The intricacy of this formulation is further enhanced when the transformation to normal coordinates is carried out.

At present no general expressions for the A coefficients are known. Quade’s transformation (21) can be used to develop a general technique for calculating the A coefficients, yet the application of the methods (a) and (b) seems to be too cumbersome in comparison with the method worked out in Section IV.


Finally, we emphasize that in an exact development of a NRLH both the kinetic and the potential energy operators must be expanded around the same nonrigid reference configuration, otherwise further approximations will be introduced and/or mathematical difficulties may arise.

VI. ON THE GEOMETRY RELAXATION

The geometry relaxation means that &+, , P:+~, . . _ , pi,+6 are allowed to vary with the effective large-amplitude coordinates in such a way that the large-amplitude internal motions take place along minimum potential energy paths, i.e., on a minimum potential energy surface, and, along these paths, the small-amplitude vibrational potential energy surface is quadratic in the first approximation of the small-amplitude curvilinear displacement representation. Then

(61)

Even if the geometry relaxation is taken into account terms linearly depending on the normal coordinates generally remain in the potential energy function, Eq. (1 l), because except for configurations corresponding to critical points on the potential energy surface the derivatives (aV/&5,),, - - - and (c?V/C%&, are not equal to zero.

Although this sounds as a general statement it can be destroyed by a special definition of the paths of large-amplitude internal motions. By the simple use of the Sayvetz conditions, as given in Eq. (lo), and Eqs. (47) and (48), one can justify the following statement. There will be no terms linearly depending on the normal coordinates in the potential energy operator if the molecular geometry is optimized along the paths of large-amplitude internal motions and if these paths satisfy the equations

aPso B~m==+m,~

0 aPL ’ (62)

where (Y = X, y, z and m = 1, 2, . . . , N for the indices. Equations (62) are related to the equations of the reaction path methods (22). A

comparison to the 2D reaction surface equations, Eqs. (2.1)-(2.6) in Ref. (22~) seems to be particularly interesting.

Since the geometry relaxation is related to certain gradients of the potential energy surface which, however, are taken into account in the derivation of the NRLH we do not need to relax (optimize) the geometry of the reference molecule. However, if the geometry is relaxed quicker convergence of the expanded Hamiltonian results.

An important fact follows from the existence of the generally nonvanishing potential energy terms depending linearly on the normal coordinates. The NRLH developed in Sections II-III can be applied only to a molecule the large-amplitude internal motions

44 VIKTOR SZALAY

of which take place on a smooth potential energy surface. Intuitively a potential energy surface can be considered to be smooth enough if

;>(g)o,. . . ,$(gJ,, (63)

where 0 denotes the average small-amplitude vibrational frequency (in cm-‘). In the case of molecules having too steep potential energy surfaces along their large-

amplitude internal coordinates the ordering given in Table I is inappropriate, and at least one modification should be made. Namely the term C, arqr should be written into the first column. Then C, arqr enters into the zero-order problem. It seems to be necessary to introduce the special large-amplitude coordinates defined by Eq. (62) for such molecules.

VII. SUMMARY AND DISCUSSION OF THE THEORY DEVELOPED IN SECTIONS II-VI

The effective rotation-large-amplitude internal motion Hamiltonian of the general molecule has been derived in two steps. In the first step the exact (within the Bom- Oppenheimer approximation) rotation-large-amplitude internal motion vibration Hamiltonian of a molecule having L large-amplitude internal motions has been expanded as a Taylor series using dimensionless normal coordinates, around a nonrigid reference configuration. Having the terms of the expanded Hamiltonian arranged by the use of the ordering scheme of Ref. (7) the method of the small-amplitude vibrational contact transformation was applied. The effective rotation-large-amplitude internal motion Hamiltonian then has been obtained as the small-amplitude vibrational average of the transformed Hamiltonian.

Since the effective Hamiltonian so derived contains isotopically dependent force constants a method had to be developed for calculating the isotopically dependent force constants from the isotopically independent ones. A general and relatively simple method has been given in Section IV. It is based on the recognition that it must be easier to get the matrix 1, and thus the tensor L, by diagonalizing an appropriate mass weighted force constant matrix than by following the NRB method (7, 8), if this force constant matrix is easy to derive. Fortunately that force constant matrix can be derived easily (see Eq. (56)). It should be noted that the result, Eq. (56), is in agreement with the result of Carrington and Miller (22a).

The method developed in Section IV assumes the knowledge of the explicit formulas of the quantities BP" and B4m,g". It is possible, however, to treat the term

; 2 %sW?s = ; 2 2 ( ~)OL?WJ~ 1: K1 Tvis (64) rs IS i

as a perturbation. Then only a few details of the effective rotation-large-amplitude motion Hamiltonian, Eq. (22), change. All the rotation-large-amplitude internal motion vibration functions remain unchanged except w: and WY changing to w:* and wt** as


The corrections of w: and w: were calculated by making use of the &, and s’& operators given in Table II. These corrections agree with the corresponding terms of the effective large-amplitude bending potential (see Ref. (23) for a correct comparison) as they are applied to triatomics. In this version of the theory the formulas of BplmsBn do not have to be known.

The use of the NRLH is, however, limited to large-amplitude internal motions which have much lower frequencies (-K’W) than the small-amplitude vibrations have. This problem can be avoided by applying a large-amplitude internal motion-small- amplitude vibration contact transformation. Some of our latest calculations show that such a contact transformation can be carried out generally. Of course there are different methods to solve this problem. One has just been developed and applied to the large- amplitude bending of triatomic molecules (23, 24).

The effect of doubly degenerate vibrations which remain degenerate during the large-amplitude internal motions is fully taken into account in the NRLH. The degeneracy, however, can be removed by the large-amplitude internal motions leading to Coriolis or Fermi resonance interactions between the two components of the orig- inally degenerate vibrations. The effect of this type of doubly degenerate vibration is only partly described by the formulas given in Eqs. (23)-(26). but it can be taken into account exactly by the direct diagonalization of an appropriate Hamiltonian matrix. A few aspects of doubly degenerate vibrations, the degeneracy of which is removed by the large-amplitude internal motions, have been discussed by Jensen (25).

Some attention should be paid to the steepness of the potential energy surface along the large-amplitude internal coordinates, for in the case of a too steep surface the original ordering scheme given in Table I can be destroyed.

Studying the question of geometry relaxation we have derived some new relations leading to a natural generalization of the 1D reaction path Hamiltonian (22~) to an LD reaction surface Hamiltonian.

VIII. APPLICATION TO C3

This section is devoted to demonstrate how the NRLH theory can be applied in practice, on the example of the C3 molecule. The results will be presented along with the corresponding results of the NRB theory.’

The general expressions of the rotation-large-amplitude internal motion vibration functions, Eqs. (23)-(36), reduce to those of Table III for triatomic molecules having a large-amplitude bending motion and the NRLH can be written as

’ The author is indebted to Dr. Philip R. Bunker for giving him the NRB results.

46 VIKTOR SZALAY

TABLE III

Explicit Formulas for the Rotation-Large-Amplitude Internal Motion Functions Building Up the NRLH of a Triatomic Molecule

r+s

r,s = 1,3

It has been discussed by Jensen and Bunker (see Ref. (8) and the references therein)

that some higher order terms of the pseudo potential energy term ULAM given in Eq. ( 13b) should be taken into account and the effective Hamiltonian should be regrouped to cope with the singularity problems arising at the linear configuration. This procedure leads to a small change in the expression of the effective bending potential. Only that

--b$hiho 5 3

FIG. 1. The numbering of the atoms and the definition of the internal coordinates.

THE NONRIGID ROTATION-LAM HAMILTONIAN

TABLE IV

The Nonzero Force “Constants” when the Potential Surface Given by Eq. (70) is Expanded According to Eq. (39)

47

f2222P b+ 1 i?! f(6)P6 +

1 k! f(8)pB + 10! f(lO)P

10

F, = fzzp + & f2222P3 + & f(fj)p5 + $- f(8)P' + & f(,0)p9 + j+ f(,2)P11

FI = F, = -flzAp + 'hf~~~Ap*

F 1

zt = f22 + - fZ2ZZP2 + - f 2 1! (6)P* + 1

f(,OjPB + 10! f(12)P 10

F 222 = f2222P +'f 3+1-f 1

3! (fJjP 5! (8)” + F f(10)P7 + k f(12)P9

F I

2222 = fP222 + ; $j)p2 + G f(8)p* +

AP = P-P,

Note. The notation F12 = (a2V/ar,aP),,=,,,~=,~,,,,, etc. is introduced.

part of U”,, which does not tend to infinite when the molecule goes to linear will contribute to the effective bending potential. This term will be denoted by o”,,,. Thus the effective bending potential of a triatomic molecule is written as

l/e,= u:+ I?.&,+ v,+ l&+0:* u1+wf*U3+X,~U,U1+X33U3U3+X~3U]U3 (68)

VIKTOR SZALAY 48

where

(69)

Let us consider the ground electronic state potential energy surface of C3 determined by ab initio calculations in Ref. (26) as a potential energy surface expressed in terms of curvilinear internal coordinates Fr, 6, and Cu corresponding to the two bond stretching and the bending motions (see also Fig. 1). Then it is written as

+ $ f;zz(Ar; + Ar;)A(u2 + f J; Izz(Af? + Ar’:)A&‘, (70)

where AS = G - c+, A& = & - r,, Af; = Q - r,, and j = ‘IT - 5.

The nonvanishing force “constants” obtained by expanding I’(&, &, ii) according to the small changes of the curvilinear internal coordinates J, , &, 5 (see Eq. (39)) are gathered into Table IV.

The expressions of the elements of tensor L for the stretching vibrations has been given up to the quartic terms in Ref. (8) and for those of the bending coordinate p one can consult Table V and Ref. (12).

TABLE V

The Elements of the L Tensor of the Bending Coordinate p

L; = -y-lx’

L rs = P

+- 1 XLpLs-y- ,.p

Lrst = P

L~;LpX(L;L;t+L;L;t+L~L;s)-y-lXrst

L rstu

= P

(L;L;L;“+L;L;L;“+LSLtLru*L~L~;s+L~~;t+L L L P P P

; ; p-px( L;L;tU+LFyt

s rtu +L L

+LsLrtu+ttLrsu+LtuLrs+LstLru+Lsurt PP PP PP PP PP PP

-L#,;L;)-y-‘XrStU

Xr = -(r,r,)-l(P~.r,+P~*r,)-X(L~r~l+L~r~l)

r,s,t,u - 1,3

Note. The terms Xq X”, and X”” can be obtained by differentiating x’ according to the normal coordinates QS, Q*, and Q,. X = cos p and y = sin p. The vectors P; and Pi are defined according to Ref. (12). L; and L; are the elements of the L tenson of internal coordinates r, and r,, respectively.

THE NONRIGID ROTATION-LAM HAMILTONIAN

TABLE VI

The Input Data for the NRLH Calculations

f22 (mdyne A)

f[6) (mdyne A)

f(,O) (mdyne A)

f 11 (mdyne A-’ )

f,,,, (mdyne A+)

f LZ (mdyne)

fl,, (mdyne)

P e 1.289664 A

ci 161.6” e

-0.0171 fzzz2 (mdyne A) 1.25

-53.6 fc8) (mdyne A) 2140

-66000 f(,2) (mdyne A) 1180000

10.924 f,,, (mdyne A-‘) -62.77

521 f 13 (mdyne A-‘) 0.0184

0.02548 f,,, (mdyne A-‘) 0.060

-0.0794 f 1122 (mdyne A-‘) -0.187

Note. These are from the CI-SDQ calculation in Table III of Ref. (25).

A computer program including the formulas of the NRLH theory developed through Sections II-VII and in the Appendix has been written to calculate the parameters of the NRLH of C3 in its ground electronic and ground stretching vibrational state. The input data of these calculations are summarized in Table VI.

The calculations led to the following interesting results. In Fig. 2 curves a and c

Y, (cm-9

2800 F

.* .*

(.. .-

‘2000.1 I ,

0.5 I 2 p badI ilrcd)

FIG. 2. The variation of the normal frequency of the symmetric stretching vibration with the effective

bending coordinate as calculated by the NRLH theory (curve a) and by the NRB theory (curve cf. Curve b shows W, as the function of the real bending coordinate.

50 VIKTOR SZALAY

FIG. 3. The variation of the normal frequency of the antisymmetric stretching vibration with the effective bending coordinate as calculated by the NRLH theory (curve a) and the NRB theory (curve c). Curve b coincides with curve a and shows wg as the function of the real bending coordinate.

show how the normal frequency of the symmetric stretching vibration w, changes with the effective bending coordinate p as calculated by the NRLH theory (curve a) and by the NRB theory (curve c). In Fig. 3 curves a and c show the same for the antisymmetric stretching vibration. The strong variation of w1 is a manifestation of the potential energy coupling of the symmetric stretching and the bending motions. There is no such coupling (due to symmetry) between the antisymmetric stretching and the bending motion. This is reflected by the shape of the function w3(p). Some

TABLE VII

The Variation of the Normal Frequencies of the Symmetric and Antisymmetric Stretching Vibrations (w, and wj) with the Bending Coordinate

NRB NRLH NRLH NRLH/only Eckart projectiona

<(t-ad)

0.2

0.4

0.6

0.8

t=-

WI(P) W,(P) W,(P) W,(P) WI(P)

1256.2 2144.5 1256.2 2144.6 1256.7

1290.1 2122.2 1290.2 2122.1 1291.9

1342.7 2085.6 1343.0 2085.3 1346.9

1409.0 2035.7 1410.1 2034.8 1417.0

1483.9 1973.7 1488.4 1971.9 1497.3 1971.9 1 1488.5 109.5 1 1971.9

%hen the expansion Eq. (39) is used with a projector matrix constructed from the Eckart

conditions instead of the Eckart-Sayvetz conditions the bending coordinate becomes

dynamically undefined and nonzero bending frequencies, wp, are obtained

(w,, We, wg have units of cm-‘).

Note. [ denotes the bending coordinate being used.

THE NONRIGlD ROTATION-LAM HAMILTONIAN 51

ikad)

FIG. 4. The effective bending potential of C, in the ground electronic and ground stretching vibrational

state: a, as calculated by the NRLH theory using the effective bending coordinate: b, as calculated by the

NRLH theory using the real bending coordinate: c, as calculated by the NRB theory as a function of the

effective bending coordinate.

numerical values of w*(p) and 03(p) are shown in Table VII, which also includes data showing what happens if only the Eckart conditions are used to construct the projector matrix P (see Eq. (55)).

TABLE VIII

The p Dependence of the Effective Bending Potential and

Its Parameters as Calculated by the NRLH Method

p/rad VO

0.00 0.0

0.20 -13.3

0.40 -15.4

0.60 49.6

0.80 174.5

1.00 324.2

1.7 -14.0 4.1 1245.9

1.6 -14.0 4.2 1256.9

1.4 -13.6 4.1 1289.6

1.2 -12.6 3.3 1339.0

0.9 -10.5 -91.1 1375.2

0.6 - 7.2 <-lo” 1171.2 1

xx w3 Xi1 X33 Xl3

VNRLH &f (P) v~;B,(P)

2154.4 -0.63 -4.00 -1.99 1690.3

2145.8 -0.62 -4.12 -1.85 1678.3 1689.8

2121.4 -0.62 -4.41 -1.52 1680.4 1691.8

2078.0 -0.66 -4.77 -1.27 1748.4 1762.8

1951.6 -0.78 -5.14 -1.47 1735.4 1892.1

1311.2 -1.60 -5.77 -3.96. -4382.0. 2039.4 I

Note. The last column includes data of the effective bending potential obtained by the NRB method. Each datum has unit of centimeter-‘.

52 VIKTOR SZALAY

-0.64 t -2

I . . .

.a -0.66 .

FIG. 5. The variation of the x, , stretching anharmonicity coefficient: a, with the effective bending coordinate; b, with the real bending coordinate.

Among all of our results, however, the funny shape of the effective bending potential (curve a in Fig. 4) is the most interesting. Some numerical data of this potential are given in Table VIII. Curves a in Figs. 5-7 show the bending dependence of the stretching anharmonicity coefficients. For comparison a few “Tn” terms (see Table IX) used in the NRB theory to build up an effective bending potential function are also calculated by the NRLH theory and presented in Table X together with their NRB values.

To understand our funny effective bending potential we recall the warning issued in Section VI: In the case of molecules having too steep potential energy surface along their large-amplitude internal coordinates the ordering given in Table I is inappropriate.

The potential surface of C3 becomes steeper and steeper along p if p increases,

X&m? 0.2 0.4 0.6 0.6 1.0

-3.9 I I I I I

b

FIG. 6. The variation of the x3, stretching anharmonicity coefficient: a, with the effective bending coordinate; b, with the real bending coordinate.

THE NONRlGID ROTATION-LAM HAMILTONIAN 53

FlG. 7. The variation ofthe xi3 stretching anharmonicity coefficient: a, with the effective bending coordinate: b, with the real bending coordinate.

resulting in a force constant a, of larger and larger absolute value (see Fig. 8) and in a larger and larger negative contribution

-$#Wi’ (from the term U,)

to the effective bending potential. Obviously, this effect must be eliminated. Putting F2 = 0 (see Table IV) a “normal” potential function could be obtained, but

of course this is not an acceptable solution to eliminate or to diminish the @, force constant. We may try to determine a reaction path-like bending coordinate as suggested

TABLE IX

Definition of the “Tn” Terms Used in the NRB Method to Build Up the Effective Bending Potential Function

54 VIKTOR SZALAY

TABLE X

The p Dependence of Some “Tn” Terms as Calculated by the NRLH and the NRB Method

T6 (cm-‘) T7 (cm-l) T8 (cm-') T9 (cm-‘) T14 (cm-') p(rad).

NRLH NRB NRLH NRB NRLH NRB NRLH NRB NRLH NRB

0.2 15.6 15.6 -1.7 -1.7 -5.0 -5.0 -6.9 -6.3 0.9 1.0

0.4 15.6 15.6 -1.7 -1.7 -5.0 -5.0 -7.0 -6.4 - 0.6 -0.7

0.6 15.6 15.6 -1.8 -1.8 -5.0 -5.0 -7.2 -6.4 - 5.7 -2.7

0.8 15.8 15.7 -1.9 -1.9 -5.2 -5.1 -7.3 -6.4 - 64.8 -5.9

1.0 16.2 15.9 -2.5 -2.1 -5.9 -5.3 -7.7 -6.4 -538.9 -12.0

in Section VI. This coordinate, however, might be seriously different from p. Even the effective bending coordinate p used so far is different (in dynamical behavior) from the real bending coordinate j and confusion may arise if it is forgotten.

Both the unusual shape of the effective bending potential function and the possibility of getting confused by bending coordinates of different physical meaning will disappear if the potential surface is expanded according to Eq. (59). Before using this expansion, however, some consideration concerning the expressions of the kinetic energy must be made.

The relation transforming the mass weighted Cartesian coordinates of the atoms of a molecule derived in the molecule fixed system MS0 to those given in the laboratory system LS,

mm ‘/*r$jm = mj/*RtS + S-‘(&$x)[m~*a~!“(~) + rnz*df$j’], (71)

0.4 0.6 0.6

l . . .

-zoo- .

.

.

FIG. 8. The 9, force constant as a function of the effective bending coordinate.


where [ stands for the large-amplitude coordinate being used, should be invertible: that is, the coordinates of the center of mass of the molecule, RkS (a = x, y, z), the Euler angles of the rotation matrix S, the large-amplitude coordinate t, and the 3N Cartesian displacement coordinates &z” should be calculable from known LS coordinates m”’ rLs

If the p:tem% energy surface of the molecule studied is approximated by Eq. (59), then, since the same coordinates must be used in formulating the expressions of the kinetic and the potential energy,

E=iJ, (72)

and the value of [ can be immediately determined from the known LS coordinates. In this case only the Eckart conditions restrict the Cartesian displacement coordinates. The Sayvetz condition does not enter as a constraint now, but is automatically satisfied. Indeed, the 3N-dimensional vector of mass weighted Cartesian displacement coordinates calculated from Eqs. (9), (71), and (72) necessarily, cannot induce any change (displacement) in the large-amplitude internal coordinate and thus it is orthogonal to the 3Ndimensional gradient vector having components

,/2 affZO mm - .

( 1 at c+,’

that is, the Sayvetz condition is satisfied. Thus all the kinetic energy formulas used in the case of the effective large-amplitude

coordinate p remain valid. We just have to keep in mind that we are dealing with the real large-amplitude internal coordinate j.

TABLE X1

The Nonzero Force “Constants” when the Potential Energy Surface Given by Eq. (70) Is Expanded According to Eq. (59)

V,(F) = ; f,F’ + 1 f,*,,ij* 4 +-f k! WC” +

F, = F, = -f12A; + %f -’ 122AP

56 VIKTOR SZALAY

TABLE XII

The j? and p Dependence of Some “Tn” Terms as Calculated by the NRLH and the NRB Method, Respectively

T T T T T7 (cm-‘) T6 (cm-‘) T8 (cm-‘) T9 (cm-') Tl4 (cm-‘)

NRB NRLH NRB NRLH NRB NRLH NRB NRLH NRB

15.6 -1.7 -1.7 -5.0 -5.0 -6.9 -6.3 1.1 1.0

15.6 -1.7 -1.7 -4.9 -5.0 -7.0 -6.4 1.0 -0.7

15.6 -1.7 -1.8 -4.9 -5.0 -7.1 -6.4 - 4.5 -2.7

15.7 -1.8 -1.9 -4.9 -5.1 -7.2 -6.4 - 9.4 -5.9

15.9 -2.0 -2.1 -5.1 -5.3 -7.2 -6.4 -15.9 -12.0

Note. .$ denotes the bending coordinate being used.

Now we can return to C3. The nonvanishing force “constants” of the expansion Eq. (59) are given in Table XI. The effective bending potential calculated by the NRLH theory using the expansion Eq. (59) is shown in Fig. 4 (curve b). Some “Tn” values are presented in Table XII and values of the terms and parameters from which the effective bending potential is built up in the NRLH theory are given in Table XIII. Curves b in Figs. 5-7 and in Figs. 2-3 show the bending dependence of the anharmonicity coefficients and of the normal frequencies of the stretching vibrations, respectively. The anharmonicity coefficients calculated at the equilibrium configuration

(x 1 1 = -0.6 cm-‘, x33 = -4.2 cm-‘, xl3 = - 1.6 cm-‘) are smaller than those calculated

TABLE XIII

The j Dependence of the Effective Bending Potential and Its Parameters as Calculated by the NRLH Method

p/r-a<

0.00

0.20

0.40

0.60

0.80

1.00 2039.41 1.0

u; Go xx **

VO LAM "* w1 w3 X11

0.0 1.7 -14.0 4.1 1245.9 2154.4 -0.63

-13.3 1.6 -14.0 4.2 1257.5 2146.0 -0.61

-15.4 1.4 -13.6 4.1 1291.2 2120.9 -0.57

49.6 1.2 -12.6 3.6 1343.5 2079.5 -0.55

174.5 0.9 -10.5 1.9 1409.6 2022.9 -0.56

1690.3

1678.7

1681.0

1751.8

1881.7

2032.5 . .

324.2 0.6 - 7.2 -1.9 1484.4 1952.2 -0.63

Note. For comparison the results of the NRB method are given in the last column. Each datum has unit of centimeter-‘.

THE NONRIGlD ROTATION-LAM HAMILTONIAN 57

by the Hoy-Mills-Strey program (xi, = 2 cm-‘, x33 = 9 cm-‘, xl3 = 13 cm-‘), from the U-SD potential energy surface (26).

It is the effective bending coordinate that the NRB method uses, yet the NRB results are much like the NRLH results obtained by using the real bending coordinate.

X. SUMMARY

With the development of the nonrigid rotation-large-amplitude internal motion Hamiltonian the accurate calculation of the rotation-large-amplitude internal motion spectra of polyatomic molecules became a realizable task. The NRLH theory allows us to use any functional representation of the potential energy surface as well as to choose among large-amplitude internal coordinates having different physical meaning (effective large-amplitude internal coordinates, real large-amplitude internal coordinates, and reaction path coordinates).

The essence of the NRLH theory can be summarized as follows: (a) Let us carry out the Hoy-Mills-Strey calculations by using the Eckart-Sayvetz conditions and approximating the potential energy surface by Eq. (39). In this case effective large- amplitude internal coordinates or reaction path (surface) coordinates are used. (b) Let us carry out the Hoy-Mills-Strey calculations by using the Eckart conditions (as if the molecule studied had only 3N - 6 - L internal degree of freedom) and approximating the potential energy surface by Eq. (59). Then, the real large-amplitude internal coordinates are used and the Sayvetz conditions are automatically satisfied.

APPENDIX A

Mass dependent potential energy-like contributions follow from the terms given in Eqs. ( 13b)-( 13e) and ( 13g) of the molecular Hamiltonian. By considering the ordering of Ref. (7) one can immediately see that energy contributions of the order of magnitude K*TV,~ or larger can result only from Eqs. (I 3b) and (13g). We can write

uo= i&Mf u:. (Al)

It goes without explanation that

642)

The energy contribution of small-amplitude vibrations can be written as

Making use of the rule for differentiation of determinants

(43)

(44)

(AS)

58 VIKTOR SZALAY

According to our ordering scheme

(A7)

thus

= Tr(h?)Tr(hF) N y,~* (A8)

(A9)

(Alo)

(Al 1)

and finally

u:= &c c {Tr(h?)Tr(@) - 2Tr(hf@)} ‘v K*T”[~,

r

Tr(@) denoting the trace of the matrix h: defined in Eq. (A7) (and in Table Ib). Ut, just like U”,,, is independent of the molecular force field and depends only

on the masses of atoms and the geometry of the nonrigid reference molecule. In fact, expanding (A 11) one can obtain

646)

where the Greek indices run over x, Y, z, p,, p2, . . . , pL and

c a;Ba:6 = C .A~~P,~~,y+A~in, (A13) I m,n

where (Y’, y’ = x, y, z,

1

2e e I m’/*aMSo UT’ @a c m -ym if ff,p,Y,E=x,Y,z

-2e,,,fm~*(da~$/dpi) if %Y=x,Y,z and

A,=a = n’m P = Pi> i= 1,2,. . . ,L (Al4)

-2m~*(a2a~~/dpidpj) if a=pi, P=Pj,

and i,j= 1,2,. . . ,L

and ~u~m,l~n is defined in Eq. (55). In deriving Eq. (A 14) straightforward generalizations of Eqs. (37)-(39) of Ref. (27) were used.

It should be noted that Eq. (A12) has been derived in a recent paper of Sarka and Bunker (28), but in a slightly different way. Assuming

LZpi (=&J = 0 (a=x,y,z,i=1,2 ,..., L) (Al5)


a simplified expression has also been given for the pseudo potential energy contribution of small-amplitude vibrations in Ref. (28). To obtain results as general as possible we have given a matrix formulation of LJt .

The potential energy surface of a freely moving molecule must be invariant under translation and rotation of the molecule as a whole. Equations (A2) and (Al 1) reveal that both UtAM and LJ!j behave as a real potential energy surface as to translation and rotation. However, they depend on the masses of atoms of the molecule unlike a real potential energy surface. So they are called pseudo potential energy terms.

Finally we note that Eq. (A2) can be rewritten into a new form by using the rule for differentiation of determinants and the equation

/.&(P”-l)J?r = &3$, = a,, 3 c416)

which may be useful in calculating U&,,:

APPENDIX B

The effective large-amplitude internal coordinates are thought to be mass dependent internal coordinates defined by the Eckart-Sayvetz conditions (8). In this Appendix the way of defining the effective large-amplitude internal coordinates introduced in Section II is discussed in some detail, and the role of the Eckart-Sayvetz conditions in defining the effective large-amplitude internal coordinates is clarified.

The Eckart-Sayvetz conditions do not tell us if the effective large-amplitude internal coordinate is a bending, a puckering, or a torsional coordinate. So, in the first step one must define the effective large-amplitude internal coordinate to be, for example, a bending coordinate of the reference molecule; that is, a phenomenological definition must be given. Without this definition we could not even set up the functions a::‘(p). The Eckart-Sayvetz conditions do not change the phenomenological definition, but affect the description of the dynamics. To clarify this point let us examine how the actual values of the effective large-amplitude internal coordinates can be obtained from the known laboratory fixed (i.e., space fixed) system of coordinates of the atoms of the real molecule.

Using the known laboratory fixed system of coordinates one can calculate the instantaneous values of the internal coordinates Pi, i = I, 2, . . . , 3N - 6. Since the internal coordinates i$, j = L + 1, L + 2, . . . , 3N - 6, of the reference molecule are known, the actual values of the small-amplitude curvilinear displacement coordinates gj,j=L+ l,L+2,..., 3N - 6, can be calculated. Making use of the Eckart- Sayvetz conditions the internal coordinates of the real and the reference molecule can be related by mass dependent equations (see, for example, Eq. (3.13) in Ref. (8)). The expansion of such equations can result in Eq. (57) on the one hand, and in the equation

(i= 1,2,. . . 9 L; j,k,I,m=L+ l,L+2,. . . ,3N-6) (Bl)

60 VIKTOR SZALAY

on the other hand. The C coefficients depend on the masses of the atoms and on the internalcoordinatespj(i=1,2 ,..., L)and$)(j=L+l,L+2 ,..., 3N-6).One can obtain the actual values of the effective large-amplitude internal coordinates using Eq. (Bl).

Since there are mass dependent coefficients in Eq. (Bl) one would say the effective large-amplitude internal coordinates are mass dependent. In the same way, and referring to Eq. (57), one would claim, however, that the pi (i = 1, 2, . . . , L) are the large- amplitude internal coordinates that depend on the atomic masses. This funny situation can be resolved only by dropping the idea of mass dependent effective, and mass dependent real, large-amplitude internal coordinates. It is the curvilinear displacements IIj=pj-pj(i= 1,2,... , L) which depend on the masses of the atoms of the molecule and are directly determined by the Eckart-Sayvetz conditions. This conclusion is in accordance with the expectation that dynamical constraints should affect dynamical quantities, e.g., force constants, accelerations, and displacements. The Eckart-Sayvetz conditions do tell us how the reference molecule follows the motion of the real one and together with the potential energy surface give the dynamical “definition” of the effective large-amplitude internal coordinates.

ACKNOWLEDGMENTS

The author thanks Dr. Philip R. Bunker for many valuable discussions and advice during the course of this work, Dr. Gabor Corradi for reading a first draft of the manuscript, and Dr. Rudolf Voszka for his generous support. The author is indebted to Drs. P. R. Bunker and J. K. G. Watson for carefully reading the manuscript and suggesting improvements. He also thanks Helen Letaif for a very careful typing of the manuscript. This paper has been completed during the author’s stay as a research associate at the Herzberg Institute of Astrophysics. He is grateful to the Herzberg Institute of Astrophysics for offering him a research associate position and to Drs. J. K. G. Watson and P. R. Bunker for their kind hospitality.

RECEIVED: June 1, 1987

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