Quantum theory and molecular spectra

Volume 52, number 3

QUANTUM

15 December 1977

CHEMICAL PHYSICS LETTERS

THEORY

L. LATHOUWERS,

AND MOLECULAR

SPECTRA

P. VAN LEWEN

Dienst Teoretische en Wiskundige Natuurkunde. University of Antwerp (RUCA). Antwerp. Belgium

and M. BOUTEN Limburgs Universitair Centrum. Diepenbeek. Belgium Received 25 July 1977

The “generator coordinate method” is Proposed as a fully quantum mechanical treatment of molecular spectra.

1. Introduction Recently the classical approach to molecular spectra, based on the Born-Oppenheimer separation of electronic and nuclear motion, has been criticised by

Woolley and Sutcliffe [ 1,2]. They argue that the molecular structure hypdthesis, i.e._, the introduction of nuclear equilibrium positions, which leads to the decomposition of the molecular energies in translational, electronic, vibrational and rotational parts has only

where x stands for all particle degrees of freedom and the “collective coordinates” or generator coordinates (CC’s) OLappear as integration variables. This continuous superposition of “intrinsic” states x(xlcf) with . weight functionsf(or) produces trial functions depending on particle coordinates only. The variationally op-

timal weight factors satisfy the integral equation

been justified at a semi-classical level. The good agree-

ment between “theory” vibration spectroscopy

and “experiment”

in rotation-

and electron scattering experi-

ments is obtained by fitting disposable parameters (bond lengths and angles, force constants, . ..) which are of a classical nature and have no place in a general quantum mechanical theory of molecular eigenstates. The molecular model has been used in nuclear

physics for the description of vibrational and rotational spectra arising from the collective motion of nucleons in the nucleus. However, the separation of different types of motion is not so c!ear cut as in molecules. As a matter of fact, redundant degrees of freedom are usually introduced to describe collective motion [3]. An attractive method to resolve this inconsistency is the so-called generator coordinate method (CCM) introduced by Griffin, Hill and Wheeler [4,5] _ It uses trial states of the form

containing nels H&P)=

the hermitean hamiltonian and overlap ker-

(x(W-flx(P),

, NdO=

(x@)lxW

- (3)

In the following we want to show that it is possible to feed back these ideas to molecular theory and obtain a fully quantum mechanical description of molecular eigenstates.

2. A generator

coordinate

trial function

for molecules

In the fned nucleus approximation, i.e., in the limit of infinite nuclear masses, eigenstates of a molecule are represented by wavefunctions of the foml [6] 439

CHEiMICAL

Volume 52. number 3

PHYSICS

(4)

cp,(rlar)MR - al Here ~&lor) is an eigenfunction of the electronic hamiltonian with electronic quantum numbers n

In this equation

the nuclear positions c~play a parametric role. They describe the geometry of the coulombic force field in which the electrons are moving. For finite nuclear masses a more reasonable mo!ecular state would be

15 December 1977

LETTERS

version of our GCM approach. Whereas in the first scheme the wavefunction factorises in an electronic and a nuclear part, the CCM states are not of the product type and hence describe a coupled motion of electrons and nuclei. It must be emphasised that, while the molecular skeleton is clearly apparent in the adiabatic wavefunction, in the GCM it serves only to construct and label the basis states and is integrated out of the final result.

3. Structure of molecular where (MRIoL)is so-me function peaked around the configuration o.‘If & is determined by minimising the energy E,(o) = of the total hamiltonian the optimum values QI~can be regarded as “the nuclear equilibrium configuration”. Although in the state x,(oo) the nuclei are no longer fixed, the electrons still move in a force field determined by the configuration ‘x0. To introduce collective motion the GCM considers the CY’Sas basis state labels and introduces superpositions of the intrinsic states (6) @Jr, R)

=jf(ol)s~,#l4

N44

da -

(7)

The weight functions are determined by solving the integral equation (2) which has the solutions f,,(a) with eigenvalues Enm The latter are upper bounds to the exact energy levels and in particular eoact 4 En0 < En(cro). In this way a “band” of states G_(r,R) is generated with fixed n (electronic quantum numbers) and different m values labelling the collective states. The important feature of the GC wavefunctions is that they have no Q!dependence, the nuclear configurations appear as integration variables only. In this way the introduction of a definite molecular structure has been avoided. The GCM contains as a limiting case what is commonly known as the adisbatic approximation. If in the intrinsic state x,(r, RIa) one clamps the nuclei, I.e., if one sets d~(Rla) = 6(R - CY),the integrations in (7) can

.

be carried out to give

rpn (MOf(R)

= ILAD6 n ’ R) ’

(8)

which is the adiabatic form of the wavefunction

Thus the adiabatic approximation 440

[6].

is a fiied nucleus

spectra in the GCM

The separation of the energy of a molecular eigenstate into translational, rotational, electronic and vibrational parts is traditionally based on the introduction of Eckart’s conditions and the Born-Oppenheimer approximation. It has been shown that the derivation of this decomposition is only justified at a semi-classical level [ 1,2]_ In fact it was first derived within the framework of the old quantum theory. In a full quantum mechanical treatment of a molecule this separation should emerge as a result of consecutive mathematical approximations rather than from a semi-classical model. In this section we aim at showing that this is feasible in the CC approach_ The tools to be used for this purpose are the so-called harmonic approximation and group theory in the space of GC’s. In order to introduce vibrational modes we develop a procedure analogous to the expansion of a potential energy surface around its equilibrium configuration. It is well-known that harmonic vibrations can be treated in the GCM by assuming a gaussian form of the overlap kernel and by expanding the ratio H/A up to second order around the minimum a0 173 A(&$) = exp[H(a,B)/~(a,13~

$(o~--P)+s@

-P)J

,

= E(q))

(9)

+ ~(6a+BSa + ZSa+A 6/3+ S@+f3&3).

Here s, A and B are 3N X 3 N matrices and 6a = c11- cro are the displacements of the CC’s from their equilibrium values. The GC integral equatidn corresponding to (9) is equivalent to a set of coupled harmonic oscillators [7] _Provided the space of CC trial functions is stable under the symmetry group of the hamiltonian, normal generator coordinates Q can be introduced making use of the point group Go which leaves the equilibrium

CHEMICAL PHYSICS LEl-rERS

Volume 52, number 3

structure ffo invariant. This transformation q = La is identical to the one which gives normal coordinates from the displacements 6R = R - o+-, (81. As a result of this change of variables the oscillators decouple. The six frequencies associated with normal coordinates of rotation and translation vanish. The remaining ones are the 3N- 6 pcsitive, non-zero eigenvalues of the matrix eigenvalue problem [7]

(10) and correspond to the normal vibrational modes of the system. As a consequence of the decoupling scheme the weight functions in the harmonic approximation facto&e 3N-6

15 December 1977

4. Conclusions

It has been shown that the generator coordinate method provides a quantum mechanical scheme for describing molecular eigenstates which is free from semi-classical contamination_ It ls a non-adiabatic, variational method which contains the familiar adiabatic approximation as a limiting case. The notion of “molecular structure”, which has served a useful purpose to visualise molecular properties, is underlying as an intermediary to construct the wavefunctiori but is integrated out of the final result. The role played by the molecular point group is conserved, and used to define the normal generator coordinates_ The decomposition of the energy in electronic, translational, vibrational and rotational terms is obtained from a purely quantum mechanical point of view.

(1 l) References

and the approximate E = E(cuO) f Euas This

eigenvalues are written as -f-Erot i- Evib -

(12)

separation is here obtained as a result of a mathematical approximation scheme in the space of GC’s rather than from semi-classical arguments involving dynamical degrees of freedom. It is noteworthy that the resulting molecular wavefunctions are not of the product form (11). Deviations from the harmonic approximation can be taken into account by considering the difference between the exact and harmonic kernels as perturbations 191. They give rise to anharmonic corrections, rotation-vibration coupling, etc.

t11 R.G. Woolley, Advan. Phys. 25 (1976) 27. 121 R.G. Woolley and B.T. Sutcliffe, Chem. Phys. Letters 45

(19771393. r31 A. Bohr and B.R. Mottelson, Nuclear structure (Benjamin, New York, 1969). [41 D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1106. [51 JJ. Griffin and J-A. Wheeler, Phys. Rev. 108 (1957) 311. FJ A. Messiah, Quantum mechanics, Vol. 2 (North-Holland, Amsterdam, 1960) p. 781. D.M. Brink and A. Weiguny, Nucl. Phys. A120 (1968)59. 171 181 E.B. Wilson Jr., J.C. Decius and PC. Cross, Molecular vibrations (McGraw-Hill, New York, 1955). PI L. Lathouwers and R.L. Lazes, 3. Phys. A10 (L977), to be published.

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