- Email: [email protected]
Methods for calculation of electronic—vibrational spectra of polyatomic molecules Part I. Overlap integrals of vibrational wave functions
Journal of
~OLECU~ STRUCTURE
Journal of Molecular Structure 328 (1994) 179-188
Methods for calculation of electronic-vibrational spectra of polyatomic molecules Part I. Overlap integrals of vibrational wave functions V.I. Baranov*, D.Yu. Zelent’sov department ~~~~ys~cs, T~miryaze~ A~~~u~t~ra~Academy, T~~ry~e~ str., 49, 1275% Moscow, Russia
Received 8 June 1994
Abstract
A new, approximate method for computing the overlap integrals of vibrational wave functions in the theory of polyatomic molecule vibronic spectra has been devised. The method is based on the quasi-orthogonality of the Dushinsky matrix as well as the reduction of the general problem to consecutively allowing for the shift of normal coordinates and their mixing by excitation of the molecule. The calculation errors are less than 5% and the speed of operation is more than two orders greater than that of the previous methods. So this method provides the possibility of computing the vibrational structure of electronic spectra in the general case without invoking additional approximations (such as ignoring the Dushinsky effect). It will also be available for solving the inverse vibronic spectroscopy problems to determine the molecular structure in excited electronic states.
1. Introduction The investigations into polyato~c molecular structure in excited electronic states appear to be particularly promising. In common with all spectral theories, in the theory of el~tro~c-~brational spectra there are two types of problems, direct and inverse. The first type of problem consists of determining spectral properties for specified molecular models and the second type consists of defining molecular structure parameters using the experimental spectral data. Despite the fact that the methods for the direct electro~c-~brational problem’s solution have been developed adequately [l], attacking the inverse problem, which *Corresponding author.
is reduced to the repeated solution of the direct their essential advancement, ones, requires primarily to increase the speed of calculations without decreasing their accuracy. Calculation of overlap integrals of the vibrational wave functions (WFs) in combi~ng electronic states (Franck-Condon factors (FCFs) and Herzberg-Teller integrals (HTIs)) is a central problem in computing the vibrational structure of electronic spectra in the adiabatic approximation because, owing to normal coordinate (NC) mixing (the Dushinsky effect [2]), these integrals are multi-dimensional. Ignoring this effect, while facilitating the computations, may introduce large errors [3]. A variety of both exact and appro~mate techniques for calculating these matrix elements (MEs)
0022-2860/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZ 0022-2860(94)08372-O
180
V.I. Baranov, D.Yu. Zelent’sovlJournal of Molecular Structure 328 (1994) 179-188
has been developed in the theory of polyatomic molecule vibronic spectra [ 1,451. Together with the developed techniques to account for vibronic interaction [ 1,5,6] and semi-empirical methods for definition of molecular model parameters in excited electronic states [ 1,7], they allow predictive calculations of vibronic spectra to be performed which are not only in qualitative but also in quantitative agreement with experimental data [1,7,8]. However, all these methods, unless the Dushinsky effect is ignored, are reduced eventually to the repeated calculation of enclosed sums (building up like 2N, where N is the number of mixing NCs). Thus while it can be said that the problem of calculation of the multi-dimensional overlap integrals is principally solved, serious difficulties occur when using these techniques for molecules with a sufficiently large number of mixing NCs (N24). In this event, the time of such integral calculation becomes intolerably large and constitutes the major portion of time expenditure in electronic-vibrational spectra computations. This problem becomes particularly pressing for analyzing spectra of complicated polyatomic molecules as well as for solving the inverse vibronic problems. In the Refs. [9- 1I], we suggested a radically new variational method for solving the direct electronic-vibrational problem. Vibrational WFs of the ground electronic state have been used as a basic set and all the originated MEs have been calculated. By this means the problem has been reduced to the construction and diagonalization of the variational matrix (VM). Its matrix elements have clear physical sense and are due to the observable spectral phenomena, which is particularly important for handling the inverse problems. Moreover, with this technique, there is no need to calculate the vibrational frequencies in excited electronic states (i.e. to solve the pure vibrational problem) as well as overlap integrals, because they are directly defined by the eigenvalues and eigenvectors of the VM. In such an approach, mathematical processing of matrices of enormous dimensions (tens of thousands) is a central problem. For its solution an algorithm based on matrix perturbation theory
has been calculated [9,11]. However, this method is not well suited for calculation of the VM eigenvectors (i.e. Franck-Condon factors) because on numerous occasions attaining sufhcient accuracy requires taking into account high-order terms in the perturbation theory series. By virtue of the non-commutativity of rotation matrices and, therefore, the ambiguity of such expressions [9], this in its turn leads to further difficulties. Thus we have pointed to the necessity of further refining of present diagonalization methods as well as the variational approach as a whole and have projected the major line of such refining
[111* Further analysis has shown that the problem of immediate FCF calculation by standard methods is closely connected with the problem of the VM diagonalization in the variational approach. This connection permits a clear understanding of the physical sense of separate stages of solution as well as the parameters used and, on these grounds, significant simplification of the solution of these problems. In consequence of this analysis we offer a new, very effective method for immediate FCF calculation particularly promising for solution of high-dimensional problems. This method enables us to significantly increase (by more than two orders) the speed of operation and considerably reduce the sensitivity of algorithm to the problem’s dimension. Two concepts play a key role in this approach, namely (a) consecutive account for shift and mixing of NCs in combining electronic states (which can always be done because of shift and rotation operators commutativity) and (b) using symmetry properties of hamiltonian and quasi-orthogonality of the Dushinsky matrix. This new method for immediate calculation of the overlap integrals of vibrational WFs in the theory of polyatomic molecule vibronic spectra is discussed in detail in the present paper. In the following Parts, II and III [12,13], we will consider, based on similar principles, a more refined method for variational solution of the electronicvibrational problem as well as the results of model calculations with the help of these methods and their comparison with those obtained by other methods.
181
V.I. Baranov, D. Yu. Zefent~ov/Jo~naf of ~ofe~~Iar ~tr~t~re 328 (1994) 1%I88
2. Method for calculation of overlap integrals 2.1. General relations Generally, multi-dimensional vibrational WFs of combining electronic states are represented as a set of ha~onic-o~llator functions {$J}, (v$): $,(Q, A) = IX, m} = (det A/fl)‘/4(2’%z!)-1’2 x exp(-$XX)H,(X)
(1)
, mN) is a set of vibrational
wheremE(m1,m2,... quantum numbers, mi=0,1,2,... N
m! E
rI
mj!,
i=l
X =
2m E fi2?, i=l
&(X)
= fiH,,&) i=l
Q;'Q
&IX; m + 1)
(n’; X’jQjX; m) = 2-‘j2Q0(da(n’;
is the ~-~mensional vector column of normalized NCs, Q. = A- Ii2 , A is the diagonal matrix of vibrational frequencies, H,,(XJ is the Hermit polynomial of mith order and N here is the total number of NCs. The expression for WFs in excited state {+‘} has similar form. Normalized NCs X and X’ are related by the general coordinate transformation (the Dushinsky equation) [l] including shift, mixing and scaling of NC& X’=XX-t”rC
(2)
where Y = Qb-‘JQ,, and IE= Qb-‘,$A are the normalized mixing matrix and shift vector; J = LhL4 is the unnormalized Dushinsky matrix; A = q’ - q is a vector of changes in natural vibrational coordinates (i.e. bond lengths and angles); L, is a matrix of vibrational forms and LP is a matrix of momenta transformation f1,141. Therefore, in the general case FCFs s(n’, m) = (n’; X’lX; m) =
like HTIs (including weight factor Qk = II”,, QF, K = CN,lki is the order of HTI) are N-dimensional. In the specific cases, considering the extent of NC mixing (the magnitude of non-diagonal elements in 9), matrix 9 may be rearranged to the block-diagonal form. By this means FCFs (Eq. (3)) and HTIs may be reduced to the product of less-dimensional integrals. Hereafter we shall denote N to be the dimension of such blocks of mixing NCs (mixing blocks) and FCFs and HTIs to be normalized for O-O transition integral: F(n’,m) = 9( n’,m)/Ft(O,O). It may be that N = 1, then we shall look at the one-dimensional mixing block (with $ii # 1). Notice that with the help of relations for MEs of an harmonic oscillator it is easy to reduce HTI to the linear combination of FCFs. In particular for N=landK=l
+,t (Q',
A’>$dQ, MQ (3)
’ From here on all quantities symbolized by capital letters should be read as denoting matrices/vectors and all operations with them should be considered as matrix ones. (- is a sign of transposition.)
+ z/t;;(n’; X’{X, m - 1)) To calculate FCFs let us take advantage generating function technique [S]:
(4) of
‘I2
F(n’, m) = exp(pATf
r?CU+ OET + 5?B+ OD) (3
where U and T are N-dimensional auxiliary variables and A = 2&X97 + I)-“5
- z
columns of
c = 2(3’f + z>-’ - z
E = 4(3Sg + I)-‘3 B = -29(39 D = -2(3S
+ I)-% + I)-‘in
+ 2~, (6)
Z is the unit matrix. In this manner to find FCF P(n’, m) we need to expand the exponent on the right-hand side of Eq. (5) as a power series and to account for all factors by the terms with T”U”‘. Although this expansion holds for all values of quanta numbers, it is of limited use because, for the total inclusion of terms with T mU ‘, we have to calculate the expansion up to the power (r + d inclusively. (The quantity
182
V.I. Baranov, D. Yu. ZeIent’sov/Journal of Molecular Structure 328 (1994) 179-188
(T= CEimi we shall call the total quantum number (TQN) of WF; it plays an important role in the suggested methods.) The number of terms in the exponent expansion increases like 5N; furthermore, account must be taken of interchanging matrix indices (associated with NC numbers) in the direct products of matrices (Eq. (6)). All this increases the complexity of the FCF calculation problem. Convenient analytic or recurrent relations for FCFs of WFs with LT,d > 1 have not been gained by virtue of the fact that expansion of exponent (5) is not simple in its structure. (Such expressions were obtained only for some particular cases [4,5].) To develop, based on Eq. (5) a calculation technique, first we shall carry out the reduction of the general case (2). By this means we simplify common expressions (Eqs. (5) and (6)) considering two particular cases and then take advantage of the peculiarities of the mixing matrix structure. 2.2. Problem’s reduction For this purpose let us introduce an ancillary set of WFs (11”). Then the desired FCFs can be expressed as follows: (n’; X’lX; m) = C(n’;
X’lX”; k”)(k”; X”IX; m)
K’ (7)
A set of NCs X” associated with the functions {$J’} is whatever we define it to be, but it is wise to determine it by either of two ways: X” = 9X x”=x+9-1K.
X’ = XN + K
(8a)
x’ = 9x1’
where the last equations can be represented as: x” = x - /$’
x’ = 9X”
(8b)
Here, K’ is nothing but the vector of NC shifts calculated in the coordinates of the ground (unprimed) state. Let us consider the familiar relation [1,14] _&L, = I. Generally, systems of the dependent natural coordinates are used to describe the molecular structure and to solve the pure vibrational problem (both in the ground and in the excited electronic states) because the molecular model
parameters corresponding to such systems have clear physical meaning as well as a variety of important features (like transferability in the series of related compounds) [l]. It is just these coordinate systems that we apply to solve the vibronic problems. However, the peculiarities of the mixing matrix 9 (and also of the shift vector K) to be considered here are independent of the natural coordinate system used, so to simplify the analysis of these peculiarities we shall consider that the system of independent natural coordinates is applied. Then L;’ = I$,, ,$’ = L,, L& = I. Writing the inverse to 9 matrix as Y-’ = Q{’ LilI$-‘Q~ = Qi’&,LbQb we get:
(9) In either case, Eq. (8a) or (8b), general NC transformation (2) is represented as the superposition of two physically and mathematically distinct transformations, i.e. pure NC shifts reflecting the change in the origin (displacement of potential surface’s minimum position) and pure mixing of NCs (variations in the directions of coordinate axes and scaling). In the former case, each of the new NCs is a function of only one old coordinate (with the same number), i.e. the corresponding equation is one-dimensional, whereas in the latter case it is a function of all old NCs (equation is multi-dimensional, but more importantly, it is homogeneous). Since not only the potential surface’s orientation as a whole changes by excitation but also the relative directions of coordinate axes (oblique-angled system of NCs is deformed), NC mixing is not pure rotation - 4 is non-orthogonal. However the actual changes in potential surfaces of polyatomic molecules are small unless there is conformation and/or isomeric change of the molecule. (This case demands special consideration [ 151.) Because of this, such transformation should be sufficiently close to the rotation, i.e. 3 is quasi-orthogonal. 2.3. Quasi-orthogonality of the mixing matrix First
let us estimate
non-orthogonality
of
183
V.I. Baranov, D. Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
unnormalized Dushinsky matrix J i.e. the magnitude of non-diagonal MEs in matrix J.!. Once again, let us consider that the system of independent natural coordinates is used (see above). In this case matrices L, and L,, are represented [ 1,141 as L = L 7’12L L = L 7-‘12L where L and L, are th4e ortbogoial Pmatrikes andU T is thi diagonal matrix of eigenvalues of kinematic coefficient matrix T : &TL, = r. Then
Substitution of Eq. (11) into Eq. (10) gives: J~~I+SS~~T’L;~I+~S
(12)
Taking into account the relationship between J and 9 it can now be obtained that 93 M Z + (X’/X)Ss, where X, X’ are typical values of frequencies in combining electronic states. Evidently, for the greatly different frequencies matrix J itself is close to the unit matrix (and so is 93) whereas significant non-diagonal MEs may appear in J only for close frequencies. Finally, it can be stated that Y3=II+
=
I - i;bTL;
(10)
Therefore, the term &5TLL (6T = T’ - T), which characterizes the changes in molecular geometry by excitation, causes the non-orthogonality of J matrix. Obviously, as only the force constant matrix U changes, matrix J is orthogonal [16], i.e. the molecular potential surface in this case undergoes only rotation and stretching along the axes, whereas changes in geometrical parameters imply changes in relative directions of coordinate axes. The MEs of T are analytic functions (though of a rather unwieldy form) [14] of the molecular geometrical parameters, namely bond lengths s and angles 8, i.e. T = T(s,B). The bulk of the dependence on bond lengths is of the form T x l/s, therefore 6T%l_!=‘-” s’ s
-=--_= s’s
AS s’s
---As 1 M -SST’ sd (11)
where Ss is a typical order of the relative changes in bond lengths (actually Ss M 0.05). The angle dependence is weaker: T a sin 19,moreover, the changes in angles are as a rule significantly smaller than those in bond lengths, so we can ignore them in this estimation. Notice that the quantity &TLL is one of the variational problem’s parameters [9,12] and its immediate calculations confirm the validity of the presented estimation.
(13)
where matrix (II,which characterizes the non-orthogonality of 9, is a small parameter (< 0.05 in actual cases). Now with the help of expansion of Eqs. (5) and (6) in powers of this parameter, we shall develop a new, very powerful method for FCF calculation. Thus in view of Eqs. (7) and (8) the general problem of FCF calculation is reduced to the cases of pure shift and pure mixing of NCs. Let us consider these cases in more detail. 2.4. Pure shift The dimension of separate blocks of Dushinsky matrix may be equal to unity. For such one-dimensional mixing blocks general transformation of NCs (2) takes the form:
where diagonal ME Yii is a parameter which indicates scaling (frequency effect). Eq. (14) is a natural generalization of the pure shift case Xi = Xi + 6.i (corresponding to Yii = 1). From the general Eq. (5), it can be shown that simple analytic expressions would hold for such one-dimensional FCFs: min(n’,m)
(GWA
= &aJ 2 x
Ek z
0
(n,
Ym-k&‘-k
_
k)!;m _ /qk,
.
.
(15)
184
V.I. Baranov, D. Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
where auxiliary quantities WY
CkDm-2k
Y, = d&7(0; X’(X; m) = 2-?m! C k=Ok!(m - 2k)! 2, = Gl(n’; X
shown (see below and/or Ref. [17]) that this matrix is orthogonal with an accuracy of 02. Thus with r=2(~~+Z)-‘~=2(9+~a-1)-1 it can be derived that
X’IX; 0) = 2%‘!
In’/21ABk ck=O k!(n’
ti -
- 2k 2k)!
(16) and therefore A=cW-Z=h(fhZ)
are used, and parameters (6) in this case are: A=-
C=y~-‘-Z=;(j-‘j-‘-Z)
(20)
Y2 - 1 E=2V”=3+9-’
9* + 1
EC--j& DE-$B=--
2rc BE----9* + 1 29K Y2 + 1
(17)
(Index i is omitted here for the sake of simplicity.) Notice that Eqs. (15)-( 17) agree with those obtained by Preem [4] at 9ii = m (i.e. Jii s 1). In the subsequent discussion a set of onedimensional FCFs is conveniently considered as matrix S(‘) with elements SA:i = (n:; XilXi; Mi). Then the total set of FCFs shows up as supermatrix S which is a direct product of all S(‘) matrices: S&S’ i=t
(‘I
(18)
2.5. Pure mixing The quasi-orthogonality of the Dushinsky matrix (13) makes it possible to put forward a new approximate method for FCF calculation. Let us invoke the perturbation theory and restrict our consideration to the fist-order terms in o. As a rule, such approximation would be quite sufficient for actual calculations. Rearranging Eq. (6) for parameters A, C and E (B = D z 0, because shifts are absent, n = 0), from Eq. (5) it follows that in the case of pure mixing E = 29’” [17], where Y is a matrix of FCFs for WFs with TQNs 0, d = 1 (i.e. m,n’= (O*,...,liy ,.., O,), i= 1,N). It can be
Although previously we have advanced the efficient method for computing the inverse to ($3 + I) matrix [17], but the special convenience of Eq. (20) in comparison with Eq. (6) is the fact that for calculating parameters A, C and E the inversion of matrix (3’9 + Z) is not warranted at all. Moreover, it can be clearly seen from Eqs. (20) and (13) that MEs of A and C are of the order of o. Thus exponent (5) expansion, correct to first order in Q, takes the form: G(U, T) = 1 + -@irET)” n=l *
+ n(oET)“-‘(FAT
+ OCU)]
(21)
In the case of total neglect of cx(9 is orthogonal, A = C = 0) only terms (OET)” will be left over in Eq. (21); i.e. FCFs will be non-zero only for WFs 1c, and $’ with identical TQNs u, d equal to n. This result is sufficiently evident, since tensor components of the specified rank are interconverted as coordinate system rotates. In the above case the products of Hermit polynomials in Eq. (1) are just such components with the rank defined by TQN, and FCFs are the coefficients of this interconversion. The inclusion of first order in a terms of the form FAT and OCU brings into existence non-zero FCFs for WFs with TQNs u and d differing by 2; but these integrals are comparatively small (of the order of CX)and in many cases may be ignored. Now the FCF totality can be split into unique blocks by the magnitude of the TQNs c, d. Each block forms a peculiar matrix V(““) with MEs
V.I. Baranov, D. Yu. Zelenr’sovlJournal of Molecular Strucrure 328 (1994) 179-188
185
V?‘lu’ = (n’;X’IX;m), where WFs @ and $’ have TGs g = C~lmi and d = Czlni respectively. (Notice that under this notation V s Y(‘l’).) These blocks have the dimension dd x d,, where d,=
(;+;-l)
=
N(N+l)...(N+/&p!
1) (22)
Performed analysis permits not only the calculation of auxiliary parameters A, C, and E but also, more importantly, simple recursions for overlap integrals to be obtained. Evidently FCFs of WFs with equal TQNs (a = d) are determined (to CK~ accuracy) only by the term (oET)b in Eq. (21). As discussed above, with the help of Eq. (5) it can easily be derived that +‘)
= ;E
(23)
For FCFs with greater values of TQN we have carried out simple recurrent relations in terms of FCFs with c = -1. For example2 for (T= 2:
+ (lj\ln)(lkllm)) +
terms with cyclic permutations of indices i,j, k. (29
(24)
‘ In the numerical notations for WFs we indicate only the nonzero vibrational quantum numbers and omit the designation for the set of NCs (Xor A”). WFs of the ground state are ket-vectors and those of the excited (primed) state are bra-vectors; i.e. I&) H Ix;O,,..., lit.. . ,O,v) and (lij E (01,. , li, . . . ,O,;X’I.
Equations for (T= 4 have similar form. Unfortunately, we have failed to obtain the total recursion that expresses the dependence on both TQN values and FCF types (determining by the number of non-zero quantum numbers mi). But when actually used, it is not warranted because there is no need to calculate FCFs for a>5 since such transitions as a rule are not observed in spectra of complicated polyatomic molecules. And for 0 = 2,3,4 the number of such types is comparatively small and recurrent equations are conveniently derived separately for each combination of types. All FCFs in the blocks with d = ITf 1 (and also d=af(2k+l),k=O,l,...) are equal to zero when a shift of NCs is absent. This property is independent of the extent of matrix 9 nonorthogonality and is associated exclusively with hamiltonian symmetry. Really, by virtue of its quadratic, a hamiltonian in the harmonic approach can cause transitions only between the states with identical parity. Taking into account Eqs. (5) and (21) for FCFs in blocks F/(‘12) and V(2Yo)the following can be
186
V.I. Baranov, D.Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
obtained (012,) = 221’2Cii
(0)lilj) = 2-‘(Cij + Cji)
(2i10) = 22”2Aii
(liljl0) = 2-‘(A,
+ Aji) (26)
Recurrent relations similar to Eqs. (24) and (25) may also be easily derived for greater values of cr,d. To illustrate for the block V(lY3)we have:
11i11j2k) (lilljlkll)
=
JZ(lillk)(olljlk) =
+
(1i11j)(o12k)
(lillk)(olljll) + C1ill/)(Olljlk) (lillj)(ollkll)
+
(27)
It is easy to verify that FCFs from the blocks with 0’ = r f 2k (k = 0,1, . . .) are of the order of o. Finally, the total FCF matrix V in this case is the direct sum of the blocks V(dla): V = $D,d y@‘,u)
(28)
i.e. it has characteristic block-band structure shown in Fig. 1, where hocks with non-zero MEs are shaded and their order in cxis indicated. Let us discuss one more important question on the orthogonality of matrix V(lV1)(and, therefore, matrix E, see Eq. (23)). We invoke the quantum mechanic identities (n;_QY;m) = S,, and CpIX’; k’)(k’; X’I = 1 which must hold for the total sets of WFs (as the eigenfunctions of the hamiltonian operator). Then c(n; K
XIX’; k’)(k’; X’IX; m) = S,,,,
(2%
Evidently this equation accounts for orthogonality of total FCF matrix V (28) and is true for all cases. Let IX;m) = Ilj), IX;n) = Iii), then from Eq. (29) we derive that C( 1ilX’; k’)(k’; A”[lj) = Sij K
(30)
for all pairs of i,j (i,j = 1, N). Now it is obvious that if 9 is orthogonal, matrix I’(“‘) will also be orthogonal, because in this case FCFs (l&Y’; k’)
Fig. 1. Structure of FCF matrix V in the case of pure mixing. Blocks VCUqd)with non-zero matrix elements are shaded and their order in a is indicated.
and (k’; X’Ilj) in Eq. (30) are non-zero for the excited state WFs IX’; k’) only of the form IX’; k’) = 11:) (see above), i.e. the identity
5t1i113)(1:11j) = sij
(31)
r=l
is fulfilled. It signifies the orthogonality of matrix V(l,l). Moreover, it can also be shown that in this case matrix V has the block-diagonal form and every block Y(b?‘) is orthogonal. In the case of non-orthogonal 9 the products of the first order in o terms (as well as the terms of higher orders, see the Fig. 1) are also included in this sum. Then from Eq. (30) we get ~(1~11~)(1~11,)
r=l
+
p(1,13i)13F11j) r=l
+ ~~[(lill:2:)(1:2~ll~~ f-1
sx N
N
N
+ (1i12F1:)(2:1111j)l +r T x
s>r t>s
r=l
X (1i1~1~1~)(1~1~1~~1j) +
**a
=
Sij
V.I. Baranov, D. Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
(32) (O(a2) signifies the terms in the order of a2 and higher.) We notice that I/(111)becomes now nonorthogonal in the order of QI’, but by virtue of the small size of cw; V(‘>‘) can still be considered as being orthogonal. (In the above discussion we restricted our consideration to the first order terms in a.) 2.6. General case Invoking Eq. (7) with an intermediate set of NCs in the Eq. (8a) or (8b) (and with A” = A’ or A” = A, respectively) we divide the problem into the two cases discussed above. Then by applying Eqs. (18) and (28), FCF matrices S and V can be easily found. It is evident that in matrix form Eq. (7) implies that total FCF matrix F is a matrix product of S and V, i.e. F = S,V
(Eq.(8a)) and F = VS, (Eq. (8b))
(33) Notice that matrix V is identical in both cases. Block-band structure of V (with respectively split matrices S, and S,) permits us to use Eq. (33) block by block. The problem is especially simplified by restricting our consideration to the zeroth order terms in (Y.Then matrix V will look like V = &, V(““), i.e. it is block-diagonal and is composed of blocks Y(“+‘) of dimension (22), which can be easily stored and multiplied. Inclusion of the first order in cx terms increases the dimensions of the blocks and thereby decreases the speed of the calculation. Nevertheless the conceptual scheme of the problem’s solution remains unchanged.
187
It is of fundamental importance that the operator Qi remains in ME with unprimed variables in order to use convenient recurrent relations like that in Eq. (4). In case of Eq. (8b) we get &ii) E (k”.7 f’lQ#;m)I X @(ICY;
=
,&;S@)
XrlQilXi; mi) @Ei+t Sci)
(35)
Integrals (ky; X: IQi [Xi; mi) are one-dimensional and can be easily worked out (see Eq. (4)). Obviously in this case, too, the problem is reduced to the product of matrices V and H(“), with the matrix V = (n’; X’lX”; k”) just the same as that in Eq. (33). Therefore the time calculation of HTIs is very nearly the same as it is in the case of FCFs. By this means, HTIs can be easily and quickly calculated with different Qi operators as well as with their combinations of any order (like Qf, QiQj, etc.). Notice that only a small number of quantities are bound to be recalculated in comparison with the FCF case. In Eq. (8a), integrals (k’; X”lQilX; m) in Eq. (34) are multi-dimensional and cannot be represented in form (35). Hence, calculation of whole FCF blocks with greater TQNs is required to use relations such as Eq. (4). This problem is far more complicated, thus the second method of choosing the intermediate set of NCs (8b) is preferable in the HTI case. It is evident that for HTIs with operator Q: it is necessary that this operator remains in an ME with primed variables in Eq. (33). In this case the intermediate set of NCs in form (8a) is preferable. So the problem of immediate calculation of overlap integrals in combining electronic states (both Frank-Condon and Herzberg-Teller types) is totally solved.
2.7 Herzberg- Teller integrals 3. Conclusion Analogous obtained:
(n’; X'lQilX;
to Eq. (7), the following can be m) =
c(n’; X’IX”; k”) W’
(k”; X"lQilX;
m)
(34)
The approximate method put forward for calculating the overlap integrals of vibrational wave functions allows the computation of the vibrational structure of polyatomic molecule electronic spectra in the adiabatic approximation without
188
V.I. Baranov, D. Yu. Zelent’sovlJournal of Molecular Structure 328 (1994) 179-188
invoking additional restrictions (such as ignoring the Dushinsky effect and/or temperature distribution of molecules). Fundamental to the method is the quasiorthogonality of the mixing matrix as well as the possibility of reducing the general coordinate transformation by excitation of molecules (the Dushinsky equation) to the consecutive stages, namely, pure shift and pure mixing of normal coordinates. Analysis of the devised calculation technique, its program realization and the results of a large number of model calculations show that this method is vastly more powerful in comparison with the other available methods. For example, its speed of operation is more than two orders greater than that of the previous methods, whereas calculation errors are no more than 5% (identical to those of the previous exact methods). These questions will be discussed in Part III in detail.
Acknowledgments The authors gratefully acknowledge partial financial support of this investigation by the International Science Foundation, Grant No. M2COOO. We also thank Professor L.A. Gribov for his helpful and important comments.
References [l] L.A. Gribov and W.J. Orville-Thomas, Theory and Methods of Calculation of Molecular Spectra, Wiley, Chichester, 1988. L.A. Gribov, V.I. Baranov and B.K. Novosadov, Methods
for Calculating Electronic-Vibrational Spectra of Polyatomic Molecules, Nauka, Moscow, 1984 (in Russian). [2] F. Dushinsky, Acta Physicochim. URSS, 7 (1937) 551. [3] V.I. Baranov and A.N. Solovyev, Opt. Spectrosk, 62 (1987) 59 (in Russian). [4] V.M. Wagner, Z. Naturforsch., Teil A, 14 (1959) 81. P.A. Preem, Trudy IFA Akad. Nauk ESSR, 16 (1961) 57 (in Russian). T.E. Sharp and H.N. Rosenstock, J. Chem. Phys., 41 (1964) 3453. WI. Baranov and L.A. Gribov, Opt. Spectrosk., 45 (1978) 463; 49 (1980) 198 (in Russian). [5] V.I. Baranov, L.A. Gribov and B.K. Novosadov, J. Mol. Struct., 70 (1981) 1. [6] V.I. Baranov and L.A. Gribov, Opt. Spectrosk., 47 (1979) 91 (in Russian). [7] V.I. Baranov and L.A. Gribov, J. Mol. Struct., 70 (1981) 31. V.I. Baranov, G.N. Ten and L.A. Gribov, J. Mol. Struct., 137 (1986) 91. [8] V.I. Baranov and A.N. Solovyev, Zh. Fiz. Khim., 59 (1985) 1720 (in Russian); Zh. Prikl. Spektrosk., 46 (1987) 69 (in Russian); Opt. Spectrosk., 62 (1987) 346 (in Russian). L.L. Gladkov and V.I. Baranov, Opt. Spektrosk., 63 (1987) 669 (in Russian). [9] V.I. Baranov and D.Yu. Zelent’sov, J. Mol. Struct., 272 (1992) 283. [lo] V.I. Baranov, L.A. Gribov and D.Yu. Zelent’sov, Zh. Strukt. Khim., 34(l) (1993) 141 (in Russian). [ll] V.I. Baranov, L.A. Gribov and D.Yu. Zelent’sov, Zh. Strukt. Khim., 34(l) (1993) 149 (in Russian). [12] V.I. Baranov, L.A. Gribov and D.Yu. Zelent’sov, J. Mol. Struct., 328 (1994) 189. [13] V.I. Baranov and D.Yu. Zelent’sov, J. Mol. Struct., 328 (1994) 199. [14] M.V. Volkenstein, L.A. Gribov, M.A. Elyashevitch and B.I. Stepanov, Vibrations of Molecules, 2nd Ed., Nauka, Moscow, 1972 (in Russian). [15] L.A. Gribov, Dokl. Akad. Nauk SSSR, 292 (1987) 1161 (in Russian). [16] L.A. Gribov and V.I. Baranov, Zh. Prikl. Spektrosk., 44 (1986) 341 (in Russian). [17] V.I. Baranov and D.Yu. Zelent’sov, Zh. Strukt. Khim., 35(2) (1994) 16 (in Russian).
~OLECU~ STRUCTURE
Journal of Molecular Structure 328 (1994) 179-188
Methods for calculation of electronic-vibrational spectra of polyatomic molecules Part I. Overlap integrals of vibrational wave functions V.I. Baranov*, D.Yu. Zelent’sov department ~~~~ys~cs, T~miryaze~ A~~~u~t~ra~Academy, T~~ry~e~ str., 49, 1275% Moscow, Russia
Received 8 June 1994
Abstract
A new, approximate method for computing the overlap integrals of vibrational wave functions in the theory of polyatomic molecule vibronic spectra has been devised. The method is based on the quasi-orthogonality of the Dushinsky matrix as well as the reduction of the general problem to consecutively allowing for the shift of normal coordinates and their mixing by excitation of the molecule. The calculation errors are less than 5% and the speed of operation is more than two orders greater than that of the previous methods. So this method provides the possibility of computing the vibrational structure of electronic spectra in the general case without invoking additional approximations (such as ignoring the Dushinsky effect). It will also be available for solving the inverse vibronic spectroscopy problems to determine the molecular structure in excited electronic states.
1. Introduction The investigations into polyato~c molecular structure in excited electronic states appear to be particularly promising. In common with all spectral theories, in the theory of el~tro~c-~brational spectra there are two types of problems, direct and inverse. The first type of problem consists of determining spectral properties for specified molecular models and the second type consists of defining molecular structure parameters using the experimental spectral data. Despite the fact that the methods for the direct electro~c-~brational problem’s solution have been developed adequately [l], attacking the inverse problem, which *Corresponding author.
is reduced to the repeated solution of the direct their essential advancement, ones, requires primarily to increase the speed of calculations without decreasing their accuracy. Calculation of overlap integrals of the vibrational wave functions (WFs) in combi~ng electronic states (Franck-Condon factors (FCFs) and Herzberg-Teller integrals (HTIs)) is a central problem in computing the vibrational structure of electronic spectra in the adiabatic approximation because, owing to normal coordinate (NC) mixing (the Dushinsky effect [2]), these integrals are multi-dimensional. Ignoring this effect, while facilitating the computations, may introduce large errors [3]. A variety of both exact and appro~mate techniques for calculating these matrix elements (MEs)
0022-2860/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZ 0022-2860(94)08372-O
180
V.I. Baranov, D.Yu. Zelent’sovlJournal of Molecular Structure 328 (1994) 179-188
has been developed in the theory of polyatomic molecule vibronic spectra [ 1,451. Together with the developed techniques to account for vibronic interaction [ 1,5,6] and semi-empirical methods for definition of molecular model parameters in excited electronic states [ 1,7], they allow predictive calculations of vibronic spectra to be performed which are not only in qualitative but also in quantitative agreement with experimental data [1,7,8]. However, all these methods, unless the Dushinsky effect is ignored, are reduced eventually to the repeated calculation of enclosed sums (building up like 2N, where N is the number of mixing NCs). Thus while it can be said that the problem of calculation of the multi-dimensional overlap integrals is principally solved, serious difficulties occur when using these techniques for molecules with a sufficiently large number of mixing NCs (N24). In this event, the time of such integral calculation becomes intolerably large and constitutes the major portion of time expenditure in electronic-vibrational spectra computations. This problem becomes particularly pressing for analyzing spectra of complicated polyatomic molecules as well as for solving the inverse vibronic problems. In the Refs. [9- 1I], we suggested a radically new variational method for solving the direct electronic-vibrational problem. Vibrational WFs of the ground electronic state have been used as a basic set and all the originated MEs have been calculated. By this means the problem has been reduced to the construction and diagonalization of the variational matrix (VM). Its matrix elements have clear physical sense and are due to the observable spectral phenomena, which is particularly important for handling the inverse problems. Moreover, with this technique, there is no need to calculate the vibrational frequencies in excited electronic states (i.e. to solve the pure vibrational problem) as well as overlap integrals, because they are directly defined by the eigenvalues and eigenvectors of the VM. In such an approach, mathematical processing of matrices of enormous dimensions (tens of thousands) is a central problem. For its solution an algorithm based on matrix perturbation theory
has been calculated [9,11]. However, this method is not well suited for calculation of the VM eigenvectors (i.e. Franck-Condon factors) because on numerous occasions attaining sufhcient accuracy requires taking into account high-order terms in the perturbation theory series. By virtue of the non-commutativity of rotation matrices and, therefore, the ambiguity of such expressions [9], this in its turn leads to further difficulties. Thus we have pointed to the necessity of further refining of present diagonalization methods as well as the variational approach as a whole and have projected the major line of such refining
[111* Further analysis has shown that the problem of immediate FCF calculation by standard methods is closely connected with the problem of the VM diagonalization in the variational approach. This connection permits a clear understanding of the physical sense of separate stages of solution as well as the parameters used and, on these grounds, significant simplification of the solution of these problems. In consequence of this analysis we offer a new, very effective method for immediate FCF calculation particularly promising for solution of high-dimensional problems. This method enables us to significantly increase (by more than two orders) the speed of operation and considerably reduce the sensitivity of algorithm to the problem’s dimension. Two concepts play a key role in this approach, namely (a) consecutive account for shift and mixing of NCs in combining electronic states (which can always be done because of shift and rotation operators commutativity) and (b) using symmetry properties of hamiltonian and quasi-orthogonality of the Dushinsky matrix. This new method for immediate calculation of the overlap integrals of vibrational WFs in the theory of polyatomic molecule vibronic spectra is discussed in detail in the present paper. In the following Parts, II and III [12,13], we will consider, based on similar principles, a more refined method for variational solution of the electronicvibrational problem as well as the results of model calculations with the help of these methods and their comparison with those obtained by other methods.
181
V.I. Baranov, D. Yu. Zefent~ov/Jo~naf of ~ofe~~Iar ~tr~t~re 328 (1994) 1%I88
2. Method for calculation of overlap integrals 2.1. General relations Generally, multi-dimensional vibrational WFs of combining electronic states are represented as a set of ha~onic-o~llator functions {$J}, (v$): $,(Q, A) = IX, m} = (det A/fl)‘/4(2’%z!)-1’2 x exp(-$XX)H,(X)
(1)
, mN) is a set of vibrational
wheremE(m1,m2,... quantum numbers, mi=0,1,2,... N
m! E
rI
mj!,
i=l
X =
2m E fi2?, i=l
&(X)
= fiH,,&) i=l
Q;'Q
&IX; m + 1)
(n’; X’jQjX; m) = 2-‘j2Q0(da(n’;
is the ~-~mensional vector column of normalized NCs, Q. = A- Ii2 , A is the diagonal matrix of vibrational frequencies, H,,(XJ is the Hermit polynomial of mith order and N here is the total number of NCs. The expression for WFs in excited state {+‘} has similar form. Normalized NCs X and X’ are related by the general coordinate transformation (the Dushinsky equation) [l] including shift, mixing and scaling of NC& X’=XX-t”rC
(2)
where Y = Qb-‘JQ,, and IE= Qb-‘,$A are the normalized mixing matrix and shift vector; J = LhL4 is the unnormalized Dushinsky matrix; A = q’ - q is a vector of changes in natural vibrational coordinates (i.e. bond lengths and angles); L, is a matrix of vibrational forms and LP is a matrix of momenta transformation f1,141. Therefore, in the general case FCFs s(n’, m) = (n’; X’lX; m) =
like HTIs (including weight factor Qk = II”,, QF, K = CN,lki is the order of HTI) are N-dimensional. In the specific cases, considering the extent of NC mixing (the magnitude of non-diagonal elements in 9), matrix 9 may be rearranged to the block-diagonal form. By this means FCFs (Eq. (3)) and HTIs may be reduced to the product of less-dimensional integrals. Hereafter we shall denote N to be the dimension of such blocks of mixing NCs (mixing blocks) and FCFs and HTIs to be normalized for O-O transition integral: F(n’,m) = 9( n’,m)/Ft(O,O). It may be that N = 1, then we shall look at the one-dimensional mixing block (with $ii # 1). Notice that with the help of relations for MEs of an harmonic oscillator it is easy to reduce HTI to the linear combination of FCFs. In particular for N=landK=l
+,t (Q',
A’>$dQ, MQ (3)
’ From here on all quantities symbolized by capital letters should be read as denoting matrices/vectors and all operations with them should be considered as matrix ones. (- is a sign of transposition.)
+ z/t;;(n’; X’{X, m - 1)) To calculate FCFs let us take advantage generating function technique [S]:
(4) of
‘I2
F(n’, m) = exp(pATf
r?CU+ OET + 5?B+ OD) (3
where U and T are N-dimensional auxiliary variables and A = 2&X97 + I)-“5
- z
columns of
c = 2(3’f + z>-’ - z
E = 4(3Sg + I)-‘3 B = -29(39 D = -2(3S
+ I)-% + I)-‘in
+ 2~, (6)
Z is the unit matrix. In this manner to find FCF P(n’, m) we need to expand the exponent on the right-hand side of Eq. (5) as a power series and to account for all factors by the terms with T”U”‘. Although this expansion holds for all values of quanta numbers, it is of limited use because, for the total inclusion of terms with T mU ‘, we have to calculate the expansion up to the power (r + d inclusively. (The quantity
182
V.I. Baranov, D. Yu. ZeIent’sov/Journal of Molecular Structure 328 (1994) 179-188
(T= CEimi we shall call the total quantum number (TQN) of WF; it plays an important role in the suggested methods.) The number of terms in the exponent expansion increases like 5N; furthermore, account must be taken of interchanging matrix indices (associated with NC numbers) in the direct products of matrices (Eq. (6)). All this increases the complexity of the FCF calculation problem. Convenient analytic or recurrent relations for FCFs of WFs with LT,d > 1 have not been gained by virtue of the fact that expansion of exponent (5) is not simple in its structure. (Such expressions were obtained only for some particular cases [4,5].) To develop, based on Eq. (5) a calculation technique, first we shall carry out the reduction of the general case (2). By this means we simplify common expressions (Eqs. (5) and (6)) considering two particular cases and then take advantage of the peculiarities of the mixing matrix structure. 2.2. Problem’s reduction For this purpose let us introduce an ancillary set of WFs (11”). Then the desired FCFs can be expressed as follows: (n’; X’lX; m) = C(n’;
X’lX”; k”)(k”; X”IX; m)
K’ (7)
A set of NCs X” associated with the functions {$J’} is whatever we define it to be, but it is wise to determine it by either of two ways: X” = 9X x”=x+9-1K.
X’ = XN + K
(8a)
x’ = 9x1’
where the last equations can be represented as: x” = x - /$’
x’ = 9X”
(8b)
Here, K’ is nothing but the vector of NC shifts calculated in the coordinates of the ground (unprimed) state. Let us consider the familiar relation [1,14] _&L, = I. Generally, systems of the dependent natural coordinates are used to describe the molecular structure and to solve the pure vibrational problem (both in the ground and in the excited electronic states) because the molecular model
parameters corresponding to such systems have clear physical meaning as well as a variety of important features (like transferability in the series of related compounds) [l]. It is just these coordinate systems that we apply to solve the vibronic problems. However, the peculiarities of the mixing matrix 9 (and also of the shift vector K) to be considered here are independent of the natural coordinate system used, so to simplify the analysis of these peculiarities we shall consider that the system of independent natural coordinates is applied. Then L;’ = I$,, ,$’ = L,, L& = I. Writing the inverse to 9 matrix as Y-’ = Q{’ LilI$-‘Q~ = Qi’&,LbQb we get:
(9) In either case, Eq. (8a) or (8b), general NC transformation (2) is represented as the superposition of two physically and mathematically distinct transformations, i.e. pure NC shifts reflecting the change in the origin (displacement of potential surface’s minimum position) and pure mixing of NCs (variations in the directions of coordinate axes and scaling). In the former case, each of the new NCs is a function of only one old coordinate (with the same number), i.e. the corresponding equation is one-dimensional, whereas in the latter case it is a function of all old NCs (equation is multi-dimensional, but more importantly, it is homogeneous). Since not only the potential surface’s orientation as a whole changes by excitation but also the relative directions of coordinate axes (oblique-angled system of NCs is deformed), NC mixing is not pure rotation - 4 is non-orthogonal. However the actual changes in potential surfaces of polyatomic molecules are small unless there is conformation and/or isomeric change of the molecule. (This case demands special consideration [ 151.) Because of this, such transformation should be sufficiently close to the rotation, i.e. 3 is quasi-orthogonal. 2.3. Quasi-orthogonality of the mixing matrix First
let us estimate
non-orthogonality
of
183
V.I. Baranov, D. Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
unnormalized Dushinsky matrix J i.e. the magnitude of non-diagonal MEs in matrix J.!. Once again, let us consider that the system of independent natural coordinates is used (see above). In this case matrices L, and L,, are represented [ 1,141 as L = L 7’12L L = L 7-‘12L where L and L, are th4e ortbogoial Pmatrikes andU T is thi diagonal matrix of eigenvalues of kinematic coefficient matrix T : &TL, = r. Then
Substitution of Eq. (11) into Eq. (10) gives: J~~I+SS~~T’L;~I+~S
(12)
Taking into account the relationship between J and 9 it can now be obtained that 93 M Z + (X’/X)Ss, where X, X’ are typical values of frequencies in combining electronic states. Evidently, for the greatly different frequencies matrix J itself is close to the unit matrix (and so is 93) whereas significant non-diagonal MEs may appear in J only for close frequencies. Finally, it can be stated that Y3=II+
=
I - i;bTL;
(10)
Therefore, the term &5TLL (6T = T’ - T), which characterizes the changes in molecular geometry by excitation, causes the non-orthogonality of J matrix. Obviously, as only the force constant matrix U changes, matrix J is orthogonal [16], i.e. the molecular potential surface in this case undergoes only rotation and stretching along the axes, whereas changes in geometrical parameters imply changes in relative directions of coordinate axes. The MEs of T are analytic functions (though of a rather unwieldy form) [14] of the molecular geometrical parameters, namely bond lengths s and angles 8, i.e. T = T(s,B). The bulk of the dependence on bond lengths is of the form T x l/s, therefore 6T%l_!=‘-” s’ s
-=--_= s’s
AS s’s
---As 1 M -SST’ sd (11)
where Ss is a typical order of the relative changes in bond lengths (actually Ss M 0.05). The angle dependence is weaker: T a sin 19,moreover, the changes in angles are as a rule significantly smaller than those in bond lengths, so we can ignore them in this estimation. Notice that the quantity &TLL is one of the variational problem’s parameters [9,12] and its immediate calculations confirm the validity of the presented estimation.
(13)
where matrix (II,which characterizes the non-orthogonality of 9, is a small parameter (< 0.05 in actual cases). Now with the help of expansion of Eqs. (5) and (6) in powers of this parameter, we shall develop a new, very powerful method for FCF calculation. Thus in view of Eqs. (7) and (8) the general problem of FCF calculation is reduced to the cases of pure shift and pure mixing of NCs. Let us consider these cases in more detail. 2.4. Pure shift The dimension of separate blocks of Dushinsky matrix may be equal to unity. For such one-dimensional mixing blocks general transformation of NCs (2) takes the form:
where diagonal ME Yii is a parameter which indicates scaling (frequency effect). Eq. (14) is a natural generalization of the pure shift case Xi = Xi + 6.i (corresponding to Yii = 1). From the general Eq. (5), it can be shown that simple analytic expressions would hold for such one-dimensional FCFs: min(n’,m)
(GWA
= &aJ 2 x
Ek z
0
(n,
Ym-k&‘-k
_
k)!;m _ /qk,
.
.
(15)
184
V.I. Baranov, D. Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
where auxiliary quantities WY
CkDm-2k
Y, = d&7(0; X’(X; m) = 2-?m! C k=Ok!(m - 2k)! 2, = Gl(n’; X
shown (see below and/or Ref. [17]) that this matrix is orthogonal with an accuracy of 02. Thus with r=2(~~+Z)-‘~=2(9+~a-1)-1 it can be derived that
X’IX; 0) = 2%‘!
In’/21ABk ck=O k!(n’
ti -
- 2k 2k)!
(16) and therefore A=cW-Z=h(fhZ)
are used, and parameters (6) in this case are: A=-
C=y~-‘-Z=;(j-‘j-‘-Z)
(20)
Y2 - 1 E=2V”=3+9-’
9* + 1
EC--j& DE-$B=--
2rc BE----9* + 1 29K Y2 + 1
(17)
(Index i is omitted here for the sake of simplicity.) Notice that Eqs. (15)-( 17) agree with those obtained by Preem [4] at 9ii = m (i.e. Jii s 1). In the subsequent discussion a set of onedimensional FCFs is conveniently considered as matrix S(‘) with elements SA:i = (n:; XilXi; Mi). Then the total set of FCFs shows up as supermatrix S which is a direct product of all S(‘) matrices: S&S’ i=t
(‘I
(18)
2.5. Pure mixing The quasi-orthogonality of the Dushinsky matrix (13) makes it possible to put forward a new approximate method for FCF calculation. Let us invoke the perturbation theory and restrict our consideration to the fist-order terms in o. As a rule, such approximation would be quite sufficient for actual calculations. Rearranging Eq. (6) for parameters A, C and E (B = D z 0, because shifts are absent, n = 0), from Eq. (5) it follows that in the case of pure mixing E = 29’” [17], where Y is a matrix of FCFs for WFs with TQNs 0, d = 1 (i.e. m,n’= (O*,...,liy ,.., O,), i= 1,N). It can be
Although previously we have advanced the efficient method for computing the inverse to ($3 + I) matrix [17], but the special convenience of Eq. (20) in comparison with Eq. (6) is the fact that for calculating parameters A, C and E the inversion of matrix (3’9 + Z) is not warranted at all. Moreover, it can be clearly seen from Eqs. (20) and (13) that MEs of A and C are of the order of o. Thus exponent (5) expansion, correct to first order in Q, takes the form: G(U, T) = 1 + -@irET)” n=l *
+ n(oET)“-‘(FAT
+ OCU)]
(21)
In the case of total neglect of cx(9 is orthogonal, A = C = 0) only terms (OET)” will be left over in Eq. (21); i.e. FCFs will be non-zero only for WFs 1c, and $’ with identical TQNs u, d equal to n. This result is sufficiently evident, since tensor components of the specified rank are interconverted as coordinate system rotates. In the above case the products of Hermit polynomials in Eq. (1) are just such components with the rank defined by TQN, and FCFs are the coefficients of this interconversion. The inclusion of first order in a terms of the form FAT and OCU brings into existence non-zero FCFs for WFs with TQNs u and d differing by 2; but these integrals are comparatively small (of the order of CX)and in many cases may be ignored. Now the FCF totality can be split into unique blocks by the magnitude of the TQNs c, d. Each block forms a peculiar matrix V(““) with MEs
V.I. Baranov, D. Yu. Zelenr’sovlJournal of Molecular Strucrure 328 (1994) 179-188
185
V?‘lu’ = (n’;X’IX;m), where WFs @ and $’ have TGs g = C~lmi and d = Czlni respectively. (Notice that under this notation V s Y(‘l’).) These blocks have the dimension dd x d,, where d,=
(;+;-l)
=
N(N+l)...(N+/&p!
1) (22)
Performed analysis permits not only the calculation of auxiliary parameters A, C, and E but also, more importantly, simple recursions for overlap integrals to be obtained. Evidently FCFs of WFs with equal TQNs (a = d) are determined (to CK~ accuracy) only by the term (oET)b in Eq. (21). As discussed above, with the help of Eq. (5) it can easily be derived that +‘)
= ;E
(23)
For FCFs with greater values of TQN we have carried out simple recurrent relations in terms of FCFs with c = -1. For example2 for (T= 2:
+ (lj\ln)(lkllm)) +
terms with cyclic permutations of indices i,j, k. (29
(24)
‘ In the numerical notations for WFs we indicate only the nonzero vibrational quantum numbers and omit the designation for the set of NCs (Xor A”). WFs of the ground state are ket-vectors and those of the excited (primed) state are bra-vectors; i.e. I&) H Ix;O,,..., lit.. . ,O,v) and (lij E (01,. , li, . . . ,O,;X’I.
Equations for (T= 4 have similar form. Unfortunately, we have failed to obtain the total recursion that expresses the dependence on both TQN values and FCF types (determining by the number of non-zero quantum numbers mi). But when actually used, it is not warranted because there is no need to calculate FCFs for a>5 since such transitions as a rule are not observed in spectra of complicated polyatomic molecules. And for 0 = 2,3,4 the number of such types is comparatively small and recurrent equations are conveniently derived separately for each combination of types. All FCFs in the blocks with d = ITf 1 (and also d=af(2k+l),k=O,l,...) are equal to zero when a shift of NCs is absent. This property is independent of the extent of matrix 9 nonorthogonality and is associated exclusively with hamiltonian symmetry. Really, by virtue of its quadratic, a hamiltonian in the harmonic approach can cause transitions only between the states with identical parity. Taking into account Eqs. (5) and (21) for FCFs in blocks F/(‘12) and V(2Yo)the following can be
186
V.I. Baranov, D.Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
obtained (012,) = 221’2Cii
(0)lilj) = 2-‘(Cij + Cji)
(2i10) = 22”2Aii
(liljl0) = 2-‘(A,
+ Aji) (26)
Recurrent relations similar to Eqs. (24) and (25) may also be easily derived for greater values of cr,d. To illustrate for the block V(lY3)we have:
11i11j2k) (lilljlkll)
=
JZ(lillk)(olljlk) =
+
(1i11j)(o12k)
(lillk)(olljll) + C1ill/)(Olljlk) (lillj)(ollkll)
+
(27)
It is easy to verify that FCFs from the blocks with 0’ = r f 2k (k = 0,1, . . .) are of the order of o. Finally, the total FCF matrix V in this case is the direct sum of the blocks V(dla): V = $D,d y@‘,u)
(28)
i.e. it has characteristic block-band structure shown in Fig. 1, where hocks with non-zero MEs are shaded and their order in cxis indicated. Let us discuss one more important question on the orthogonality of matrix V(lV1)(and, therefore, matrix E, see Eq. (23)). We invoke the quantum mechanic identities (n;_QY;m) = S,, and CpIX’; k’)(k’; X’I = 1 which must hold for the total sets of WFs (as the eigenfunctions of the hamiltonian operator). Then c(n; K
XIX’; k’)(k’; X’IX; m) = S,,,,
(2%
Evidently this equation accounts for orthogonality of total FCF matrix V (28) and is true for all cases. Let IX;m) = Ilj), IX;n) = Iii), then from Eq. (29) we derive that C( 1ilX’; k’)(k’; A”[lj) = Sij K
(30)
for all pairs of i,j (i,j = 1, N). Now it is obvious that if 9 is orthogonal, matrix I’(“‘) will also be orthogonal, because in this case FCFs (l&Y’; k’)
Fig. 1. Structure of FCF matrix V in the case of pure mixing. Blocks VCUqd)with non-zero matrix elements are shaded and their order in a is indicated.
and (k’; X’Ilj) in Eq. (30) are non-zero for the excited state WFs IX’; k’) only of the form IX’; k’) = 11:) (see above), i.e. the identity
5t1i113)(1:11j) = sij
(31)
r=l
is fulfilled. It signifies the orthogonality of matrix V(l,l). Moreover, it can also be shown that in this case matrix V has the block-diagonal form and every block Y(b?‘) is orthogonal. In the case of non-orthogonal 9 the products of the first order in o terms (as well as the terms of higher orders, see the Fig. 1) are also included in this sum. Then from Eq. (30) we get ~(1~11~)(1~11,)
r=l
+
p(1,13i)13F11j) r=l
+ ~~[(lill:2:)(1:2~ll~~ f-1
sx N
N
N
+ (1i12F1:)(2:1111j)l +r T x
s>r t>s
r=l
X (1i1~1~1~)(1~1~1~~1j) +
**a
=
Sij
V.I. Baranov, D. Yu. Zelent’sov/Journal of Molecular Structure 328 (1994) 179-188
(32) (O(a2) signifies the terms in the order of a2 and higher.) We notice that I/(111)becomes now nonorthogonal in the order of QI’, but by virtue of the small size of cw; V(‘>‘) can still be considered as being orthogonal. (In the above discussion we restricted our consideration to the first order terms in a.) 2.6. General case Invoking Eq. (7) with an intermediate set of NCs in the Eq. (8a) or (8b) (and with A” = A’ or A” = A, respectively) we divide the problem into the two cases discussed above. Then by applying Eqs. (18) and (28), FCF matrices S and V can be easily found. It is evident that in matrix form Eq. (7) implies that total FCF matrix F is a matrix product of S and V, i.e. F = S,V
(Eq.(8a)) and F = VS, (Eq. (8b))
(33) Notice that matrix V is identical in both cases. Block-band structure of V (with respectively split matrices S, and S,) permits us to use Eq. (33) block by block. The problem is especially simplified by restricting our consideration to the zeroth order terms in (Y.Then matrix V will look like V = &, V(““), i.e. it is block-diagonal and is composed of blocks Y(“+‘) of dimension (22), which can be easily stored and multiplied. Inclusion of the first order in cx terms increases the dimensions of the blocks and thereby decreases the speed of the calculation. Nevertheless the conceptual scheme of the problem’s solution remains unchanged.
187
It is of fundamental importance that the operator Qi remains in ME with unprimed variables in order to use convenient recurrent relations like that in Eq. (4). In case of Eq. (8b) we get &ii) E (k”.7 f’lQ#;m)I X @(ICY;
=
,&;S@)
XrlQilXi; mi) @Ei+t Sci)
(35)
Integrals (ky; X: IQi [Xi; mi) are one-dimensional and can be easily worked out (see Eq. (4)). Obviously in this case, too, the problem is reduced to the product of matrices V and H(“), with the matrix V = (n’; X’lX”; k”) just the same as that in Eq. (33). Therefore the time calculation of HTIs is very nearly the same as it is in the case of FCFs. By this means, HTIs can be easily and quickly calculated with different Qi operators as well as with their combinations of any order (like Qf, QiQj, etc.). Notice that only a small number of quantities are bound to be recalculated in comparison with the FCF case. In Eq. (8a), integrals (k’; X”lQilX; m) in Eq. (34) are multi-dimensional and cannot be represented in form (35). Hence, calculation of whole FCF blocks with greater TQNs is required to use relations such as Eq. (4). This problem is far more complicated, thus the second method of choosing the intermediate set of NCs (8b) is preferable in the HTI case. It is evident that for HTIs with operator Q: it is necessary that this operator remains in an ME with primed variables in Eq. (33). In this case the intermediate set of NCs in form (8a) is preferable. So the problem of immediate calculation of overlap integrals in combining electronic states (both Frank-Condon and Herzberg-Teller types) is totally solved.
2.7 Herzberg- Teller integrals 3. Conclusion Analogous obtained:
(n’; X'lQilX;
to Eq. (7), the following can be m) =
c(n’; X’IX”; k”) W’
(k”; X"lQilX;
m)
(34)
The approximate method put forward for calculating the overlap integrals of vibrational wave functions allows the computation of the vibrational structure of polyatomic molecule electronic spectra in the adiabatic approximation without
188
V.I. Baranov, D. Yu. Zelent’sovlJournal of Molecular Structure 328 (1994) 179-188
invoking additional restrictions (such as ignoring the Dushinsky effect and/or temperature distribution of molecules). Fundamental to the method is the quasiorthogonality of the mixing matrix as well as the possibility of reducing the general coordinate transformation by excitation of molecules (the Dushinsky equation) to the consecutive stages, namely, pure shift and pure mixing of normal coordinates. Analysis of the devised calculation technique, its program realization and the results of a large number of model calculations show that this method is vastly more powerful in comparison with the other available methods. For example, its speed of operation is more than two orders greater than that of the previous methods, whereas calculation errors are no more than 5% (identical to those of the previous exact methods). These questions will be discussed in Part III in detail.
Acknowledgments The authors gratefully acknowledge partial financial support of this investigation by the International Science Foundation, Grant No. M2COOO. We also thank Professor L.A. Gribov for his helpful and important comments.
References [l] L.A. Gribov and W.J. Orville-Thomas, Theory and Methods of Calculation of Molecular Spectra, Wiley, Chichester, 1988. L.A. Gribov, V.I. Baranov and B.K. Novosadov, Methods
for Calculating Electronic-Vibrational Spectra of Polyatomic Molecules, Nauka, Moscow, 1984 (in Russian). [2] F. Dushinsky, Acta Physicochim. URSS, 7 (1937) 551. [3] V.I. Baranov and A.N. Solovyev, Opt. Spectrosk, 62 (1987) 59 (in Russian). [4] V.M. Wagner, Z. Naturforsch., Teil A, 14 (1959) 81. P.A. Preem, Trudy IFA Akad. Nauk ESSR, 16 (1961) 57 (in Russian). T.E. Sharp and H.N. Rosenstock, J. Chem. Phys., 41 (1964) 3453. WI. Baranov and L.A. Gribov, Opt. Spectrosk., 45 (1978) 463; 49 (1980) 198 (in Russian). [5] V.I. Baranov, L.A. Gribov and B.K. Novosadov, J. Mol. Struct., 70 (1981) 1. [6] V.I. Baranov and L.A. Gribov, Opt. Spectrosk., 47 (1979) 91 (in Russian). [7] V.I. Baranov and L.A. Gribov, J. Mol. Struct., 70 (1981) 31. V.I. Baranov, G.N. Ten and L.A. Gribov, J. Mol. Struct., 137 (1986) 91. [8] V.I. Baranov and A.N. Solovyev, Zh. Fiz. Khim., 59 (1985) 1720 (in Russian); Zh. Prikl. Spektrosk., 46 (1987) 69 (in Russian); Opt. Spectrosk., 62 (1987) 346 (in Russian). L.L. Gladkov and V.I. Baranov, Opt. Spektrosk., 63 (1987) 669 (in Russian). [9] V.I. Baranov and D.Yu. Zelent’sov, J. Mol. Struct., 272 (1992) 283. [lo] V.I. Baranov, L.A. Gribov and D.Yu. Zelent’sov, Zh. Strukt. Khim., 34(l) (1993) 141 (in Russian). [ll] V.I. Baranov, L.A. Gribov and D.Yu. Zelent’sov, Zh. Strukt. Khim., 34(l) (1993) 149 (in Russian). [12] V.I. Baranov, L.A. Gribov and D.Yu. Zelent’sov, J. Mol. Struct., 328 (1994) 189. [13] V.I. Baranov and D.Yu. Zelent’sov, J. Mol. Struct., 328 (1994) 199. [14] M.V. Volkenstein, L.A. Gribov, M.A. Elyashevitch and B.I. Stepanov, Vibrations of Molecules, 2nd Ed., Nauka, Moscow, 1972 (in Russian). [15] L.A. Gribov, Dokl. Akad. Nauk SSSR, 292 (1987) 1161 (in Russian). [16] L.A. Gribov and V.I. Baranov, Zh. Prikl. Spektrosk., 44 (1986) 341 (in Russian). [17] V.I. Baranov and D.Yu. Zelent’sov, Zh. Strukt. Khim., 35(2) (1994) 16 (in Russian).