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Elimination of asymptotic couplings in molecular dynamics
27 January 1995
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical PhysicsLetters 232 (1995) 457-462
,,
Elimination of asymptotic couplings in molecular dynamics A. Riera Departamento de Qufmica, C-IX, Universidad Aut6noma de Madrid. Cantoblanco, 28049 Madrid. Spain
Received 25 July 1994; in final form 22 November1994
Abstract Two successful procedures that eliminate asymptotic couplings in close-coupling treatments of atomic dynamics are shown to be generalizable to systems with three or more atoms. The procedures are easily implemented, and employ either common generalized coordinates or common electron translation factors.
In close-coupling treatments involving several electronic wavefunctions, there can arise asymptotic couplings between these functions which are a hindrance when imposing the boundary conditions for vibrorotational as well as dissociation problems. In the latter problems, those couplings become important at nuclear velocities v > 0.1 au; accordingly, their existence was first recognized [1], and they have been thoroughly discussed, in the treatment of atomic collisions, where energies are usually high enough to excite the electronic state of the reagents. By now, the problem, and its solutions, have become standard (see reviews in Refs. [2-6] ) in this field. In particular, most (semiclassical) treatments working at nuclear velocities that are lower than the electronic ones have employed the so-called common translation factor (CTF) method [7]. This method provides an efficient procedure that is nearly as easy to implement as the original closecoupling one, for molecular as well as atomic basis sets [6,8]. The present Letter considers, in general terms and keeping the algebra as simple as possible, whether the experience reached in atomic collisions can be generalized to the more general field of molecular dynamics, in either a quantal or semi-classical context. Atomic units will be used throughout.
We start by describing the asymptotic couplings. As will be clear in the following, the theory applies to time-dependent as well as time-independent formalisms, to any number of nuclei, and to any kind of close-coupling basis functions presenting residual couplings. To be specific, however, we first consider a time-independent treatment, and a basis of wavefunctions ~ , , representing a triatomic molecule ABC in the Born-Oppenheimer approximation, '/'~ = g~j ( RA, RB, Rc) ~b.j (r; RA, RB, Rc ),
( 1)
where r stands for the set of N electronic variables (including spin) of the system and the nuclear coordinates RA, RB, Rc are defined with respect to a common origin (._9of the molecule, such as the centre of mass; X~j is a nuclear function, and &i the electronic one, fulfilling He1 (bj(r; RA, RB, RC) = E j ( R A , RB, Rc) q~.j(r; RA, RB, Rc),
(2)
where Hel is the fixed-nuclei Hamiltonian, depending parametrically upon RA, RB, Rc. Close-coupling treatments make use of the completeness of the solutions of Eq. (2) for each value
0009-2614/95/$09.50 © 1995 Elsevier ScienceB.V. All rights reserved. SSD10009-2614(94)01389-6
458
A. Riera / Chemical Physics Letters 232 (1995) 45 7-462
of RA, Ra and Rc, and expand the total wavefunction in terms of a set of functions of the form ( 1 ). Upon substitution in the Schr6dinger equation this yields a system of coupled differential equations for the nuclear functions X~j, to be solved with the appropiate boundary conditions. We now consider the situation where a nucleus A, carrying with it NA electrons, is far away from the rest BC; this can correspond to either a local vibrational mode or a dissociation coordinate,
~ = x A(RA )g~,jBC(RB, Rc)
× qbj(r; RA, R B , R c )
(3)
and &.i is an antisymmetrized product of atomic (A) and molecular (BC) electronic wavefunctions. Then, the nuclear kinetic operator produces asymptotic nonadiabatic couplings, e.g., between the atomic wavefunction of A and the infinite set of those that are dipole-coupled to it. This follows from the identity
PRA ]rj = --P ]"ARB"C+ PRA IrAj'
(4)
with p the total electronic linear momentum operator, PR^ the corresponding one for nucleus A, which is taken by keeping fixed NA electronic coordinates with respect to the center (..9 in the left-hand side, while they are kept fixed with respect to the nucleus in the last term of (4). A similar relation applies to the other fragment BC. The last term of (4) acting upon an electronic wavefunction of A yields zero when the jth electron belongs to this atom; however, the preceeding term does not, and gives rise to residual couplings. Obviously, residual couplings such as those just described arise because the wavefunctions (1) cannot satisfy the total (nuclei + electrons) Schr6dinger equation in the asymptotic region. They reflect an intrinsic limitation of the electronic component &j to dynamically describe the dissociation ABC--* A + BC. To see this, we consider a limit behaviour of the nuclear function X,~ of (3) (assuming for simplicity that at least one of the two A and BC are not ionic), as RA --~ Oo,
A ( RA ) --~ RA 1 exp(iqAjRA)T(RA), Xaj
(5)
where T is an (internal) rotational function whose specific form is irrelevant in the present context, and q,Aj is the limit value for the local momentum,
A 2 (q~aC)2 (qaj) + _ _ , E~ - Ej = 2MA 2MBc
(6)
with MA and MBC the masses of the dissociation fragments. When RA describes vibration, E,~ < Ej in the limit, and q,Aj is imaginary. When RA describes dissociation, E,~ > Ej, q~Ai is real, and (5) is an (outgoing) dissociative wave. In either case, the expression (5) is multiplied in (3) by the electronic wavefunction ~bj (r; RA, RB, Rc ), which has been calculated for fixed nuclei, and cannot describe the following by the electronic cloud of the nuclear motion of A. This is most easily seen when q,Ai is real, and the velocity of --1 A the electron cloud of A should be equal to M A qaj. However, since 4/ is (usually) real, its current density, hence its overall velocity, vanishes. Hence, Eq. (1) (or (3)) cannot fulfill the Schr6dinger equation in the limit, and the couplings that arise through the operator /~A • P IRA in (4) are a consequence of this fact [9,10]. To eliminate asymptotic couplings, a modification of ( 1 ) is necessary, which should describe, in particular, the following of the nucleus of A by ?CA electrons in the limit RA ~ oc, with similar changes for other dissociation limits. This can be done in several ways, However, to be really useful, a solution to the problem should satisfy the following conditions: (I) It should eliminate the asymptotic couplings by using a workable formalism. For most applications in molecular dynamics and spectroscopy, it is amply sufficient that the boundary conditions be satisfied to first order in the inverse nuclear masses. This approximation considerably simplifies the calculations, and will be employed throughout this Letter. (II) It should be compatible with any symmetry property of the total wavefunction. In particular, the antisymmetry with respect to electron exchange, fulfilled by ( 1), should be preserved. (III) An advantage of (super)molecular closecoupling treatments is that a time-consuming calculation of the electronic Hamiltonian matrix elements is avoided, by using Eq. (2). A desideratum is that the improvement of the expansion does not destroy this property. (IV) It is also desirable that the ultimate justification for close-coupling expansions, that is, their convergence as the basis becomes complete, is not impaired by the modifications. One thus ensures that im-
459
A. Riera / Chemical Physics Letters 232 (1995) 457-462
proving the asymptotic behaviour does not worsen the treatment elsewhere. In a recent work [ 10], a remedy to the so-called infrared paradox [ 1 1-13 ] was proposed to account for the dynamics of electron clouds in the vibro-rotational movement of (diatomic) molecules, and based on an application of the method of generalized coordinates [ 1 4 - 1 6 ] , developed in the field of atomic collisions. Since this technique solves the problem of asymptotic couplings in both collisional and vibro-rotational issues, an extension to polyatomics would then permit to solve this problem in the general case of molecular dynamics. We now show that such a generalization is straightforward. To lowest order in the inverse nuclear masses, the generalized coordinate approach method consists in substituting in the nuclear function of Eq. (1) the variables Rj by Aj ( J =A, B, C),
the caret denoting a unit vector. This is by no means the only possibility, and the conclusion from the recent review of Ref. [6] is that its precise analytical expression is not crucial in the energy range which is of interest here. Hence, it may be constructed such that the calculation of the new matrix elements it gives rise to be as easy as possible. One should only ensure that the corrections introduced in the couplings are small [6] for all distances RA, RB, RC of importance in the dynamics. The substitution Rj :=~ ~j solves the asymptotic problem to first order in the inverse nuclear masses. For example, in the limit of Eq. (3) it yields . A ~A A)Xct.j " BC'A t[tct =Xc~jt ~ B, AC)
x ¢bj(r;RA
× ~ b j ( r ; RA = A A , R B = A B , R c = A C ) ,
(10)
where from (5) we have, to first order in MA 1, x~Aj(AA)
l/tf~ -- ,)t%'j ( AA, AB, AC )
= AA,RB = AB,Rc = AC),
(7) x exp
* g A' exp(iqAigA) . A lqc~j
f(
7'(RA )
ri, RA ) RA " ri
•
( 1]
i=1
with N
Aj = Rj 4-Z
f(ri,
(8)
Rj)ri,
i=1
A ( AA ) -~ R A 1 exp(iqAiRA) T(RA) X~j
where ~bj is evaluated for values of RA, RB and Rc that are numerically equal to AA, AB and Ac- f is the so-called switching function [7], which for a polyatomic we can define: in the limit RA --+ ~ (for example), when an electron remains bound to nucleus A, so that r tends to RA, we must have f(ri, RA) --+ MA I, where MA may be taken to be the nuclear mass (the difference between nuclear and atomic masses giving rise to higher order corrections that are irrelevant in the present approximation); in the opposite limit direction, when r --~ --RA, which corresponds to the electron remaining bound to the other fragment BC (the origin O lying in between of A and BC), f(ri, RA) -+ 0. In other words, switching functions provide smooth interpolations between the possible asymptotic limit situations for the electron. They are also easy to construct. As an illustration, a possible form is f ( r i, R j ) = ( 2 M j ) - 1 (
From the properties of the switching function, when NA electrons remain bound to nucleus A,
1
4- Rj
* !')
(9)
x exp
iM q~j ~ ~A
"
r,.
.
(12)
When multiplied by the atomic function of A, this yields the correct asymptotic form (to first order in M21 ) for the NA electrons to follow the nucleus. It may be readily seen that for a diatomic AB we obtain
AA =R+Z[f(ri,
A = AB
--
RB)-- f(ri, RA)]ri,
(13)
i
which coincides, to order M j j ( J = A, B), with that proposed in Ref. [ 10]. It should be stressed that use of the new closecoupling basis functions (7), rather than ( 1), only requires modest changes in the programming and computational efforts. For this, one may employ, in the
460
A. Riera / Chemical Physics Letters 232 (1995) 45 7-462
set of coupled equations defining the nuclear functions X~j, the relation between the usual electronic momentum operator Pi IR^RBgc (with RA, RB, Rc and rj withj 4: i fixed) and the new onePi [aAaBac (with AA, ~a, AC and rj with j 4= i fixed),
Pi iSARaRc = Pi lagaaac
Z
+
[Vif(ri, R j ) . r i + f(ri, Ra)] Paxlr,r s,
J=A,B,C
(14) with Par the generalized 'nuclear' momenta. The last sum in Eq. (14) gives rise to corrections in the nonadiabatic couplings, written in terms of one-electron o p e r a t o r s ~ r i f and f ; these corrections result in that all modified couplings vanish in the asymptotic region [ 10]. Moreover, in the resulting matrix elements, to an excellent approximation Pa~bj can be replaced by PRjq~j and (~j (r; RA = ,AA,RB = ~tB, RC = /tC) by ~bj (r; RA, RB, Rc) ; also, new terms of order M~"2 ( J = A, B, C) may often be neglected. If we now consider again the region RA ~ c~, we see that the effect of the coordinate ~tA is to multiply the electronic wavefunction in (10) by a translation factor (the last factor in ( 11 ) and (12) ). Furthermore, when Ej/E~ << 1 in Eq. (6), the value ofqAj becomes independent of the electronic state, and we can use an average quantity qA ~ qaj. A Within this approximation, and provided that asymptotic couplings in the treatment of vibration are not a difficulty, a formalism that is asymptotically equivalent to the use of generalized coordinates consists in modifying the wavefunction of ( 1 ) by a CTF exp(iU), for all values of RA, RB, Rc,
ltrta = Xaj(RA, RB, Rc) exp[iU(r; RA, RB, Rc)] × q~/(r; RA, RB, Rc),
(15)
with N
U= ~ J=A,B,C
qJ~ - ' f ( r i , Rj)Ra.ri.
(16)
i=l
Alternatively, any other CTF with the same asymptotic form may be adopted. As is the case with the wavefunctions (7), the use of (15) as a close-coupling basis only requires modest changes in the computational effort. Thus, when the form of the switching function is sensibly chosen, the calculation of the correc-
tive couplings, which are integrals involving the oneelectron functions Vi f and f , may be easier than that of the usual, dynamical ones, while modifications in the electronic energies are usually negligible at very low nuclear velocities. We now briefly consider a time-dependent treatment of the dynamical problem, in which each nucleus J asymptotically moves with a velocity vj. It may then be seen [ 5,7 ] that the corresponding asymptotic couplings are eliminated by multiplication of the electronic wavefunction &j [r; RA(t), RB(t), Rc(t) ], for all values of RA, RB, Rc, by a CTF exp(iU), where U is now written, N
U=
Z J=A,B,C
MJZf[ri'Rj(t)]vj'ri"
(17)
i=l
Again, any function U that asymptotically tends to (17) will also do. Incidentally, it may be mentioned that for presentation reasons the definition of switching function (here and in Ref. [10]) is slightly different from the usual one in atomic dynamics, where with only two nuclei it is simpler to define f to vary between the limits - 1 / 2 and 1/2, so that m j f would be replaced by f in Eq. (17). It is easy to see that both common coordinates AA, ~B, Ac, and CTFs exp(iU) satisfy the conditions ( I ) - ( I V ) . For example, they are symmetric with respect to electron exchange, which guarantees fulfillment of condition (II) with respect to antisymmetrization. Likewise, condition (III) is satisfied, and the completeness of the set {~/',~} in ( 1 ) is preserved in (7) and (15) [2,3], so that (IV) is fulfilled. As shown in Ref. [6], the use of a common factor exp(iU) is not physically restrictive, since it yields widely dissimilar average velocities for the electron clouds corresponding to different electronic states. Furthermore, the CTF-modified electronic wavefunction exp(iU)~bj can be expressed in the formalism of the so-called traveling orbitals. In this formalism, one takes into account that ~bj is usually written in terms of configurations built from a set of atomic or molecular orbitals {~Pn}. One can then replace these 'static' orbitals by 'traveling' ones, (Pn(ri) ~ ~On(ri) exp
(iwjn • ri)
,
(18)
where the w/, are chosen so that the boundary conditions are satisfied; for instance, wj, can be propor-
A. Riera /Chemical Physics Letters 232 (1995) 45 7-462
tional to a switching function, or a (constant or R~dependent, and j-dependent or j-independent) nuclear velocity. In the particular case
w.jn= Z
q~f(ri, R.,)Rj
J=A,B,C
= ~
Mjf(ri, Rj)vj,
(19)
J=A,B,C
with f the switching function of (16) and (17), we see that all the phases in (18) factor out from ~bj, so that the procedure is equivalent to multiplying ~bj by the CTF exp(iU) with U given by (16) and (17). The formalism of traveling orbitals (18) also permits to incorporate non-common translation factors in the electronic wavefunction. However, this procedure has not been employed for systems with more than one active electron, and is computationally awkward: (i) there arise momentum transfer integrals, containing factors of the type exp(iAk • ~-~jrj) in the integrand, with A k a momentum transfer vector; even by expanding this phase in powers of ]A kl, the calculation of these integrals can lengthen the calculation; (ii) condition (III) above is not fulfilled, so that most of the work performed when solving Eq. (2) is wasted when the resulting wavefunctions ~i are modified, with the result that couplings due to Hel need to be evaluated. Moreover, condition (IV) is not fulfilled either, which indicates that the modified closecoupling expansion may in fact be worsened (save in the asymptotic region), rather than improved [ 17], by using (18) to eliminate the residual couplings. This last point was specifically studied in Ref. [ 18] for the simplest non-common choice of wi,, which is, for an atomic orbital g,, centered on nucleus J ( J = A, B, C), to employ the limit value, J M j- I R j = vj, Win = qc~
(20)
so that residual couplings are eliminated. The solution (18) and (20) was proposed in Refs. [19-21], and is related to the pioneer work of Bates and McCarroll [ 1 ], as well as to the use of atomic close-coupling expansions, for which it has been successfully employed at high and intermediate velocities. Nevertheless, unlike (18) and (20) the method of Ref. [ 1 ] is inapplicable to molecular expansions for systems with more than one electron [6], and for a single electron both
461
methods do not coincide. Furthermore, it was shown in Ref. [ 18] that, in the region where the electronic wavefunctions {~bj} take up a delocalized structure, they are so severely distorted at velocities c > 0. I au by the substitution (18) and (20) that they lose their usefulness as a close-coupling basis. The distortion is smaller the lower the velocity; however, even at very low velocities it does not seem practical to implement a computationally difficult method that eliminates asymptotic couplings at the expense of slightly worsening the expansion elsewhere - especially since the simpler and better procedure of using a common factor is available. To sum up, two procedures that have emerged as most useful to eliminate asymptotic couplings in treatments of atomic collisions at low nuclear velocities have been shown to be generalizable to molecular dynamics. Common generalized coordinates (8) provide the general solution that is advocated here for quantum mechanical treatments of collisional and vibro-rotational problems. Alternatively, common translation factors (16) may be used, provided the total energy E, is high enough, and the residual couplings in vibration are not a problem. In the form (17), CTFs provide the general solution advocated here tot approaches where the nuclei follow classical trajectories. Generalized coordinates and CTFs have been defined in the present work in terms of switching functions f(ri, Rj), which is certainly not the only possibility, although problably the simplest one. The precise form of f(ri, Rj) is to be chosen so that calculation of corrective terms is amenable to a fast calculation [6]. This calculation does not require the evaluation of momentum transfer integrals [22], and can be easily implemented in any close-coupling treatment. While wavefunctions modified by a CTF can be written in terms of traveling orbitals, the use of such orbitals involving non-common translation factors does not provide a competitive procedure for molecular bases. This research has been partially supported by the DGICYT project No. PB93-0288-C02.
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A. Riera / Chemical Physics Letters 232 (1995) 45 7-462
References [ 1] D.R. Bates and R. McCarroll, Proc. Roy. Soc. A 245 (1958) 175. [2] B.C. Garrett and D.G. Truhlar, Theoret. Chem. Advan. Perspectives Part A 6 (1981) 215. [3] J.B. Delos, Rev. Mod. Phys. 53 (1981) 287. [4] A. Macfas and A. Riera, Phys. Rept. 90 (1982) 299. [51 B.H. Bransden and M.R.C. McDowell, Charge exchange and the theory of ion-atom collisions (Clarendon Press, Oxford, 1992). [6] L.E Errea, C. Harel, H. Jouin, L. M6ndez, B. Pons and A. Riera, J. Phys. B 27 (1994) 3603. 171 S.B. Schneiderman and A. Russek, Phys. Rev. 181 (1969) 1311. 18 [ L.E Errea, C. Harel, H. Jouin, L. M6ndez, B. Pons and A. Riera, Phys. Rev. A 50 (1994) 418. [9] A. Riera, J. Mol. Struct. THEOCHEM 300 (1993) 93.
[10] A. Riera, J. Chem. Phys. 99 (1993) 2891. I l l ] N.V. Cohan and H.E Hameka, J. Chem. Phys. 45 (1966) 4392. [12] T.H. Walnut and L.A. Nafie, J. Chem. Phys. 67 (1977) 1491. [13] Y. Mar6chal, J. Chem. Phys. 84 (1985) 247. [14] M.H. Mittleman, Phys. Rev. 188 (1969) 231. [151 M.H. Mittleman and H. Tai, Phys. Rev. A 8 (1973) 1880. [161 T.A. Green, Phys. Rev. A 23 (1981) 519. [171 W.R. Thorson and J.B. Delos, Phys. Rev. A 18 (1978) 117. ll8] L.E Errea, L. M6ndez and A. Riera, J. Phys. B 15 1982) 101. [19] A. Riera and A. Salin, J. Phys. B 9 (1976) 2877. [201 J.B. Delos and W.R. Thorson, J. Chem. Phys. 70 1979) 1774. [21] L.E Errea, L. M6ndez and A. Riera, J. Phys. B 12 1979) 69. 122] E. Deumens, A. Diz, R. Longo and Y. Ohm, Rev. Mod. Phys. 66 (1994) 917.
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical PhysicsLetters 232 (1995) 457-462
,,
Elimination of asymptotic couplings in molecular dynamics A. Riera Departamento de Qufmica, C-IX, Universidad Aut6noma de Madrid. Cantoblanco, 28049 Madrid. Spain
Received 25 July 1994; in final form 22 November1994
Abstract Two successful procedures that eliminate asymptotic couplings in close-coupling treatments of atomic dynamics are shown to be generalizable to systems with three or more atoms. The procedures are easily implemented, and employ either common generalized coordinates or common electron translation factors.
In close-coupling treatments involving several electronic wavefunctions, there can arise asymptotic couplings between these functions which are a hindrance when imposing the boundary conditions for vibrorotational as well as dissociation problems. In the latter problems, those couplings become important at nuclear velocities v > 0.1 au; accordingly, their existence was first recognized [1], and they have been thoroughly discussed, in the treatment of atomic collisions, where energies are usually high enough to excite the electronic state of the reagents. By now, the problem, and its solutions, have become standard (see reviews in Refs. [2-6] ) in this field. In particular, most (semiclassical) treatments working at nuclear velocities that are lower than the electronic ones have employed the so-called common translation factor (CTF) method [7]. This method provides an efficient procedure that is nearly as easy to implement as the original closecoupling one, for molecular as well as atomic basis sets [6,8]. The present Letter considers, in general terms and keeping the algebra as simple as possible, whether the experience reached in atomic collisions can be generalized to the more general field of molecular dynamics, in either a quantal or semi-classical context. Atomic units will be used throughout.
We start by describing the asymptotic couplings. As will be clear in the following, the theory applies to time-dependent as well as time-independent formalisms, to any number of nuclei, and to any kind of close-coupling basis functions presenting residual couplings. To be specific, however, we first consider a time-independent treatment, and a basis of wavefunctions ~ , , representing a triatomic molecule ABC in the Born-Oppenheimer approximation, '/'~ = g~j ( RA, RB, Rc) ~b.j (r; RA, RB, Rc ),
( 1)
where r stands for the set of N electronic variables (including spin) of the system and the nuclear coordinates RA, RB, Rc are defined with respect to a common origin (._9of the molecule, such as the centre of mass; X~j is a nuclear function, and &i the electronic one, fulfilling He1 (bj(r; RA, RB, RC) = E j ( R A , RB, Rc) q~.j(r; RA, RB, Rc),
(2)
where Hel is the fixed-nuclei Hamiltonian, depending parametrically upon RA, RB, Rc. Close-coupling treatments make use of the completeness of the solutions of Eq. (2) for each value
0009-2614/95/$09.50 © 1995 Elsevier ScienceB.V. All rights reserved. SSD10009-2614(94)01389-6
458
A. Riera / Chemical Physics Letters 232 (1995) 45 7-462
of RA, Ra and Rc, and expand the total wavefunction in terms of a set of functions of the form ( 1 ). Upon substitution in the Schr6dinger equation this yields a system of coupled differential equations for the nuclear functions X~j, to be solved with the appropiate boundary conditions. We now consider the situation where a nucleus A, carrying with it NA electrons, is far away from the rest BC; this can correspond to either a local vibrational mode or a dissociation coordinate,
~ = x A(RA )g~,jBC(RB, Rc)
× qbj(r; RA, R B , R c )
(3)
and &.i is an antisymmetrized product of atomic (A) and molecular (BC) electronic wavefunctions. Then, the nuclear kinetic operator produces asymptotic nonadiabatic couplings, e.g., between the atomic wavefunction of A and the infinite set of those that are dipole-coupled to it. This follows from the identity
PRA ]rj = --P ]"ARB"C+ PRA IrAj'
(4)
with p the total electronic linear momentum operator, PR^ the corresponding one for nucleus A, which is taken by keeping fixed NA electronic coordinates with respect to the center (..9 in the left-hand side, while they are kept fixed with respect to the nucleus in the last term of (4). A similar relation applies to the other fragment BC. The last term of (4) acting upon an electronic wavefunction of A yields zero when the jth electron belongs to this atom; however, the preceeding term does not, and gives rise to residual couplings. Obviously, residual couplings such as those just described arise because the wavefunctions (1) cannot satisfy the total (nuclei + electrons) Schr6dinger equation in the asymptotic region. They reflect an intrinsic limitation of the electronic component &j to dynamically describe the dissociation ABC--* A + BC. To see this, we consider a limit behaviour of the nuclear function X,~ of (3) (assuming for simplicity that at least one of the two A and BC are not ionic), as RA --~ Oo,
A ( RA ) --~ RA 1 exp(iqAjRA)T(RA), Xaj
(5)
where T is an (internal) rotational function whose specific form is irrelevant in the present context, and q,Aj is the limit value for the local momentum,
A 2 (q~aC)2 (qaj) + _ _ , E~ - Ej = 2MA 2MBc
(6)
with MA and MBC the masses of the dissociation fragments. When RA describes vibration, E,~ < Ej in the limit, and q,Aj is imaginary. When RA describes dissociation, E,~ > Ej, q~Ai is real, and (5) is an (outgoing) dissociative wave. In either case, the expression (5) is multiplied in (3) by the electronic wavefunction ~bj (r; RA, RB, Rc ), which has been calculated for fixed nuclei, and cannot describe the following by the electronic cloud of the nuclear motion of A. This is most easily seen when q,Ai is real, and the velocity of --1 A the electron cloud of A should be equal to M A qaj. However, since 4/ is (usually) real, its current density, hence its overall velocity, vanishes. Hence, Eq. (1) (or (3)) cannot fulfill the Schr6dinger equation in the limit, and the couplings that arise through the operator /~A • P IRA in (4) are a consequence of this fact [9,10]. To eliminate asymptotic couplings, a modification of ( 1 ) is necessary, which should describe, in particular, the following of the nucleus of A by ?CA electrons in the limit RA ~ oc, with similar changes for other dissociation limits. This can be done in several ways, However, to be really useful, a solution to the problem should satisfy the following conditions: (I) It should eliminate the asymptotic couplings by using a workable formalism. For most applications in molecular dynamics and spectroscopy, it is amply sufficient that the boundary conditions be satisfied to first order in the inverse nuclear masses. This approximation considerably simplifies the calculations, and will be employed throughout this Letter. (II) It should be compatible with any symmetry property of the total wavefunction. In particular, the antisymmetry with respect to electron exchange, fulfilled by ( 1), should be preserved. (III) An advantage of (super)molecular closecoupling treatments is that a time-consuming calculation of the electronic Hamiltonian matrix elements is avoided, by using Eq. (2). A desideratum is that the improvement of the expansion does not destroy this property. (IV) It is also desirable that the ultimate justification for close-coupling expansions, that is, their convergence as the basis becomes complete, is not impaired by the modifications. One thus ensures that im-
459
A. Riera / Chemical Physics Letters 232 (1995) 457-462
proving the asymptotic behaviour does not worsen the treatment elsewhere. In a recent work [ 10], a remedy to the so-called infrared paradox [ 1 1-13 ] was proposed to account for the dynamics of electron clouds in the vibro-rotational movement of (diatomic) molecules, and based on an application of the method of generalized coordinates [ 1 4 - 1 6 ] , developed in the field of atomic collisions. Since this technique solves the problem of asymptotic couplings in both collisional and vibro-rotational issues, an extension to polyatomics would then permit to solve this problem in the general case of molecular dynamics. We now show that such a generalization is straightforward. To lowest order in the inverse nuclear masses, the generalized coordinate approach method consists in substituting in the nuclear function of Eq. (1) the variables Rj by Aj ( J =A, B, C),
the caret denoting a unit vector. This is by no means the only possibility, and the conclusion from the recent review of Ref. [6] is that its precise analytical expression is not crucial in the energy range which is of interest here. Hence, it may be constructed such that the calculation of the new matrix elements it gives rise to be as easy as possible. One should only ensure that the corrections introduced in the couplings are small [6] for all distances RA, RB, RC of importance in the dynamics. The substitution Rj :=~ ~j solves the asymptotic problem to first order in the inverse nuclear masses. For example, in the limit of Eq. (3) it yields . A ~A A)Xct.j " BC'A t[tct =Xc~jt ~ B, AC)
x ¢bj(r;RA
× ~ b j ( r ; RA = A A , R B = A B , R c = A C ) ,
(10)
where from (5) we have, to first order in MA 1, x~Aj(AA)
l/tf~ -- ,)t%'j ( AA, AB, AC )
= AA,RB = AB,Rc = AC),
(7) x exp
* g A' exp(iqAigA) . A lqc~j
f(
7'(RA )
ri, RA ) RA " ri
•
( 1]
i=1
with N
Aj = Rj 4-Z
f(ri,
(8)
Rj)ri,
i=1
A ( AA ) -~ R A 1 exp(iqAiRA) T(RA) X~j
where ~bj is evaluated for values of RA, RB and Rc that are numerically equal to AA, AB and Ac- f is the so-called switching function [7], which for a polyatomic we can define: in the limit RA --+ ~ (for example), when an electron remains bound to nucleus A, so that r tends to RA, we must have f(ri, RA) --+ MA I, where MA may be taken to be the nuclear mass (the difference between nuclear and atomic masses giving rise to higher order corrections that are irrelevant in the present approximation); in the opposite limit direction, when r --~ --RA, which corresponds to the electron remaining bound to the other fragment BC (the origin O lying in between of A and BC), f(ri, RA) -+ 0. In other words, switching functions provide smooth interpolations between the possible asymptotic limit situations for the electron. They are also easy to construct. As an illustration, a possible form is f ( r i, R j ) = ( 2 M j ) - 1 (
From the properties of the switching function, when NA electrons remain bound to nucleus A,
1
4- Rj
* !')
(9)
x exp
iM q~j ~ ~A
"
r,.
.
(12)
When multiplied by the atomic function of A, this yields the correct asymptotic form (to first order in M21 ) for the NA electrons to follow the nucleus. It may be readily seen that for a diatomic AB we obtain
AA =R+Z[f(ri,
A = AB
--
RB)-- f(ri, RA)]ri,
(13)
i
which coincides, to order M j j ( J = A, B), with that proposed in Ref. [ 10]. It should be stressed that use of the new closecoupling basis functions (7), rather than ( 1), only requires modest changes in the programming and computational efforts. For this, one may employ, in the
460
A. Riera / Chemical Physics Letters 232 (1995) 45 7-462
set of coupled equations defining the nuclear functions X~j, the relation between the usual electronic momentum operator Pi IR^RBgc (with RA, RB, Rc and rj withj 4: i fixed) and the new onePi [aAaBac (with AA, ~a, AC and rj with j 4= i fixed),
Pi iSARaRc = Pi lagaaac
Z
+
[Vif(ri, R j ) . r i + f(ri, Ra)] Paxlr,r s,
J=A,B,C
(14) with Par the generalized 'nuclear' momenta. The last sum in Eq. (14) gives rise to corrections in the nonadiabatic couplings, written in terms of one-electron o p e r a t o r s ~ r i f and f ; these corrections result in that all modified couplings vanish in the asymptotic region [ 10]. Moreover, in the resulting matrix elements, to an excellent approximation Pa~bj can be replaced by PRjq~j and (~j (r; RA = ,AA,RB = ~tB, RC = /tC) by ~bj (r; RA, RB, Rc) ; also, new terms of order M~"2 ( J = A, B, C) may often be neglected. If we now consider again the region RA ~ c~, we see that the effect of the coordinate ~tA is to multiply the electronic wavefunction in (10) by a translation factor (the last factor in ( 11 ) and (12) ). Furthermore, when Ej/E~ << 1 in Eq. (6), the value ofqAj becomes independent of the electronic state, and we can use an average quantity qA ~ qaj. A Within this approximation, and provided that asymptotic couplings in the treatment of vibration are not a difficulty, a formalism that is asymptotically equivalent to the use of generalized coordinates consists in modifying the wavefunction of ( 1 ) by a CTF exp(iU), for all values of RA, RB, Rc,
ltrta = Xaj(RA, RB, Rc) exp[iU(r; RA, RB, Rc)] × q~/(r; RA, RB, Rc),
(15)
with N
U= ~ J=A,B,C
qJ~ - ' f ( r i , Rj)Ra.ri.
(16)
i=l
Alternatively, any other CTF with the same asymptotic form may be adopted. As is the case with the wavefunctions (7), the use of (15) as a close-coupling basis only requires modest changes in the computational effort. Thus, when the form of the switching function is sensibly chosen, the calculation of the correc-
tive couplings, which are integrals involving the oneelectron functions Vi f and f , may be easier than that of the usual, dynamical ones, while modifications in the electronic energies are usually negligible at very low nuclear velocities. We now briefly consider a time-dependent treatment of the dynamical problem, in which each nucleus J asymptotically moves with a velocity vj. It may then be seen [ 5,7 ] that the corresponding asymptotic couplings are eliminated by multiplication of the electronic wavefunction &j [r; RA(t), RB(t), Rc(t) ], for all values of RA, RB, Rc, by a CTF exp(iU), where U is now written, N
U=
Z J=A,B,C
MJZf[ri'Rj(t)]vj'ri"
(17)
i=l
Again, any function U that asymptotically tends to (17) will also do. Incidentally, it may be mentioned that for presentation reasons the definition of switching function (here and in Ref. [10]) is slightly different from the usual one in atomic dynamics, where with only two nuclei it is simpler to define f to vary between the limits - 1 / 2 and 1/2, so that m j f would be replaced by f in Eq. (17). It is easy to see that both common coordinates AA, ~B, Ac, and CTFs exp(iU) satisfy the conditions ( I ) - ( I V ) . For example, they are symmetric with respect to electron exchange, which guarantees fulfillment of condition (II) with respect to antisymmetrization. Likewise, condition (III) is satisfied, and the completeness of the set {~/',~} in ( 1 ) is preserved in (7) and (15) [2,3], so that (IV) is fulfilled. As shown in Ref. [6], the use of a common factor exp(iU) is not physically restrictive, since it yields widely dissimilar average velocities for the electron clouds corresponding to different electronic states. Furthermore, the CTF-modified electronic wavefunction exp(iU)~bj can be expressed in the formalism of the so-called traveling orbitals. In this formalism, one takes into account that ~bj is usually written in terms of configurations built from a set of atomic or molecular orbitals {~Pn}. One can then replace these 'static' orbitals by 'traveling' ones, (Pn(ri) ~ ~On(ri) exp
(iwjn • ri)
,
(18)
where the w/, are chosen so that the boundary conditions are satisfied; for instance, wj, can be propor-
A. Riera /Chemical Physics Letters 232 (1995) 45 7-462
tional to a switching function, or a (constant or R~dependent, and j-dependent or j-independent) nuclear velocity. In the particular case
w.jn= Z
q~f(ri, R.,)Rj
J=A,B,C
= ~
Mjf(ri, Rj)vj,
(19)
J=A,B,C
with f the switching function of (16) and (17), we see that all the phases in (18) factor out from ~bj, so that the procedure is equivalent to multiplying ~bj by the CTF exp(iU) with U given by (16) and (17). The formalism of traveling orbitals (18) also permits to incorporate non-common translation factors in the electronic wavefunction. However, this procedure has not been employed for systems with more than one active electron, and is computationally awkward: (i) there arise momentum transfer integrals, containing factors of the type exp(iAk • ~-~jrj) in the integrand, with A k a momentum transfer vector; even by expanding this phase in powers of ]A kl, the calculation of these integrals can lengthen the calculation; (ii) condition (III) above is not fulfilled, so that most of the work performed when solving Eq. (2) is wasted when the resulting wavefunctions ~i are modified, with the result that couplings due to Hel need to be evaluated. Moreover, condition (IV) is not fulfilled either, which indicates that the modified closecoupling expansion may in fact be worsened (save in the asymptotic region), rather than improved [ 17], by using (18) to eliminate the residual couplings. This last point was specifically studied in Ref. [ 18] for the simplest non-common choice of wi,, which is, for an atomic orbital g,, centered on nucleus J ( J = A, B, C), to employ the limit value, J M j- I R j = vj, Win = qc~
(20)
so that residual couplings are eliminated. The solution (18) and (20) was proposed in Refs. [19-21], and is related to the pioneer work of Bates and McCarroll [ 1 ], as well as to the use of atomic close-coupling expansions, for which it has been successfully employed at high and intermediate velocities. Nevertheless, unlike (18) and (20) the method of Ref. [ 1 ] is inapplicable to molecular expansions for systems with more than one electron [6], and for a single electron both
461
methods do not coincide. Furthermore, it was shown in Ref. [ 18] that, in the region where the electronic wavefunctions {~bj} take up a delocalized structure, they are so severely distorted at velocities c > 0. I au by the substitution (18) and (20) that they lose their usefulness as a close-coupling basis. The distortion is smaller the lower the velocity; however, even at very low velocities it does not seem practical to implement a computationally difficult method that eliminates asymptotic couplings at the expense of slightly worsening the expansion elsewhere - especially since the simpler and better procedure of using a common factor is available. To sum up, two procedures that have emerged as most useful to eliminate asymptotic couplings in treatments of atomic collisions at low nuclear velocities have been shown to be generalizable to molecular dynamics. Common generalized coordinates (8) provide the general solution that is advocated here for quantum mechanical treatments of collisional and vibro-rotational problems. Alternatively, common translation factors (16) may be used, provided the total energy E, is high enough, and the residual couplings in vibration are not a problem. In the form (17), CTFs provide the general solution advocated here tot approaches where the nuclei follow classical trajectories. Generalized coordinates and CTFs have been defined in the present work in terms of switching functions f(ri, Rj), which is certainly not the only possibility, although problably the simplest one. The precise form of f(ri, Rj) is to be chosen so that calculation of corrective terms is amenable to a fast calculation [6]. This calculation does not require the evaluation of momentum transfer integrals [22], and can be easily implemented in any close-coupling treatment. While wavefunctions modified by a CTF can be written in terms of traveling orbitals, the use of such orbitals involving non-common translation factors does not provide a competitive procedure for molecular bases. This research has been partially supported by the DGICYT project No. PB93-0288-C02.
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A. Riera / Chemical Physics Letters 232 (1995) 45 7-462
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