Born-Oppenheimer and Pseudo-Jahn-Teller Effects as considered in the Framework of the Time-Dependent Adiabatic Approximation An Intuitive Approach B. Boulil, 0. Henri-Rwsseau, and M. Deumie Universite de Perpignan, Avenue de Villeneuve. 66025 Perpignan, France It is well known that the Born-Oppenheimer approximation (.I . is the usual framework for theseparation ofelectronic from nuclear motions. At a deeper level, the Born-Oppenheimer approximation is considered as a special case of what is commonly called the "adiabatic approximation" (2). I t must he shown that this approximation does not involve any time dependence, hut rather matrix elements that take into account thr conjugate momentum uf the vibrational roordinntcs. For sim~licitvweshall call t h i s a ~ ~ r o x i m a t i oROAA. n - On the other hand, the term "adiabstic" applies in quantum mechanics to an a~oroximation (3) that is used when .. the Hamiltmian R o f asysrem changesslowlyn~ithtimp. It is of interest to note here that this apwoximation states that a system defined at any time by an kigenfunction of A may be considered as remaining in this state longer and longer as A changes more and more slowly with time. In order to distinguish this last approximation from the one above, we shall call i t the time-dependent adiabatic approximation, TDAA. I t is obvious that the BOAA is aconsequence of the TDAA since its physical basis is that the nuclei may be considered classically as moving slowly with respect to the electrons. However, i t must be noted that the link between these two approximations is not explicitly given in textbooks, since it is difficult to treat i t in a riaorous and straiehtforward fashion. Keverrheless, we thinkihat the questic;n is of sufficient interest to be considered, even if this must be done inruitive-
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IY.
The purpose of this paper is, therefore, to show that neglecting the jumping probability in the TDAA is equivalent to ignoring the matrix elements that are neglected in the BOAA. We shall also try to show how the pseudo-JahnTeller effect (4) is linked to the TDAA when the jumping probability cannot he further ignored. The Born-Oppenheimer Adlabatlc Approxlmation Let us start by a short review of the BOAA. We shall first express the matrix elements that appear in the BOAA. T o simplify, we shall limit our discussion in this section to the diatomic HT ion. In the absence of an external field and disregarding the translation of the mass center of the molecule and the rotation, we obtain for the molecular Hamiltonian
where h is the Planck constant, M the reduced mass of the molecule, m the mass of the electron, Q the nuclear coordinate. 4 the set of electronic coordinates, and U(q, Q) the operator. Then, the Schrodinger eqnation for the system as a whole
Recall that the common typing for the wavefunction is Wq, Q); however, we think that it is better to note WQ). The first step in the BOAA is to solve the Schrodinger equation for the studied case, which is identical with eq 2 except that the kinetic energy operator of the nuclear coordinates is omitted; that is, we solve for the motion of the electrons in the potential energy of interactions involving Coulomb attraction and repulsion, hut holding the nuclear position fixed. Consequently, we have the following Schrodinger equation = Ern(Q)[d9)I(~l H"(Q)[dq)li~~
(3)
where Ho(Q)is given by
and where E,(Q) and [rp,(q)](Q, are, respectively, the eigenvalue and the eigenvector of the Hamiltonian H o . Here, we prefer the more explicit notation [rp,(q)](~)to the usual writingq,(q, Q) in order to indicate that for eachvalue of Q there is a function v,(q). I t must be underlined that [q,(q)ltg) depends on Q as a parameter and so does the energy E,(Q), where [ ~ , ( q ) ] ( ~ ) depends on q as a variable. Next, according to the BOAA we use E A Q ) as a potential energy function to discuss the nuclear motion. That is, we have to solve a Schrodinger eqnation of the form
When the two Schrodinger equations (eqs 3 and 5) have been solved, one for the electronic motion and the other for the nuclear, then the BOAA states that the energy r,j of eq 5 forms a good approximation to the energy levels of the exact Schrodinger eqnation (eq 2). Furthermore, it states that a good approximation to the wave function WQ) of the exact problem is provided by the product
Now, to solve the exact Schrodinger equation (eq 21, it is possible to use, as basis for describingR, the set of wavefunctions [rp,(q,Q)~,(Q)) and to write WQ) =
1[ ' P ~ ( ~ ) I ~ Q I [ X ~ ~ ~ ~ Q I
(6)
rnj
Consequently, the average value of the Hamiltonian A on the expanded wavefunction Wq, Q) given by eq 6 will appear to be amatrix: the matrix elements will be given by multiplying one term bithe sum appearing in the r&ht sidp of eq ti hy the operator R a n d premultiplying rhe result by the product ~
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May 1988
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which, by changing with time, will induce the variation of H, will be the nuclear coordinate Q. Keeping that in mind, we may observe from eq 13 that the change of the electronic hamiltonian Hothrough the Q variation induces some probability of transition from the initial electronic state ,l)' to another one b,), and that the probability amplitude will decrease if the variation of Hoin time gets slower. In the limit where the Ehrenfest condition is satisfied the system will remain in the initial state Irp,) when Q is changing. This is nothing else but the BOAA, which allows us to construct correlation diagrams and to consider given electronic states while disregarding the nuclear displacements. As it may he observed, it is not clear that eq 13 involves implicitly coupling terms analogous to that of eq 10. Consequently, we shall show now that neglecting the jumping probability given by eq 13 is equivalent to neglecting the matrix elements given in eq 11. First, it may be observed that the change with time of the Hamiltonian H is performed through the variations of the Q coordinates on which H" is depending. Therefore, we may write
back and forth. Consequently, the parameter of interest will he the average value of C,,(t) given by eq 16 on aperiod T of oscillation; that is,
C,, = T1SIC,,(t)Idt However, since the nuclei of the molecule obey also the quantum mechanics, it is preferable to calculate the quantum average of C,, in place of the time average. As a consequence, we have to consider C,,(t) as an operator C,,(Q) only depending on the Q coordinate and to express the classical velocity dQldt in a quantum mechanical fashion. We have therefore to pass from the velocity dQldt to the impulsion P through dQ - P dt M
and then consider P as a quantum operator which is given, according to quantum mechanics, by
This leads to the transformation of eq 16 into the following operator equation:
This leads to express the partial derivative of H with respect to time by
and to make for the eigenstates and the eigenvalues of Ha, the following change in notation I&)) = lai[Q(t)l) = 14,~) Ei(t) = E,IQ(t)l = E;(Q)
where the index i stands for n and m. Now, let us neglect the fast change with time of the phase factor appearing in eq 13. Then, using the above equations, we obtain from eq 13 h [('"l%~qm)(Q)} C,, " : r [EJQ) - E,(Q)I~ It,
.(9)
Then, we have to calculate the average value of C,,(Q) on the vibrational wave function characterizing the nuclei movement. Of course, we must consider two vibrational wavefunctions, one x,+ associated to the final electronic state I*,) and the other one, x,, associated to the initial electronic state Ip,). This leads to the following mean transition value for C,,(t) between the initial ,y,k vibrational state and the final one x,,j:
or in the Dirac formalism (14)
dt
where the label (t) denotes the time dependence of the Q coordinate. At this step we have to make some transformation (6). First, we may write
Moreover, as we have from eq 5
Using eq 17, we obtain, therefore,
Now, to proceed further in thediscussion, we have to make the followine assumption, which consists of taking, for the difference (i -n E&), some average value (En- E m ) , as is usually done, and then writing
H~I'P,)(Q) = E~(Q)IP~)IQ)
we obtain
As a consequence of the above equation, eq 14 becomes
The BOAA as a Consequence of the TDAA In the Ehrenlest Approxlmatlon: Average Value of the Transition Probability
For the diatomic molecule case discussed here we may make the assumption that the nuclei are classically moving
Moreover, let us denote
so that eq 19 becomes
At this step, we may observe that (Cnm)kjhas the form of a first-order Rayleigh-Schrodinger perturbation coefficient mixing the perturbed state b,)lx,j) with the other one Ip,)lXnk). By comparing the above result with eq 12, we find that is precisely one of the expressed coefficients of this equation since eq 20 is the same as eq 11. Volume 65
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The Pseudo-Jahn-Teller Effect and the Born-Oppenheimer Approximation as Depending on the The-Dependent Adiabatic Approximation Hm,m
p=--IH-,-,
P
- H+
