Bom-Oppenheimer separation

The neglected part of the molecular Schroedinger equation, after making the Bom-Oppenheimer separ ation in the first section of this chapter, is... 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

Correlation rules relate the symmetry of reactants to the symmetry of products. More precisely, they give the symmetry of the fragments which can result when a molecule or transition state is distorted in the direction of reactants or products32,33. A familiar example is the correlation of the states of a diatomic molecule with those of its constituent atoms. Within the Bom-Oppenheimer separation we can deal with strictly electronic correlation rules, valid when there is negligible coupling between electronic and vibrational wave functions. When such coupling is important, correlations forbidden on a strictly electronic basis may be allowed, so the validity of purely electronic correlation rules is hard to assess for polyatomic molecules with strongly excited vibration. 

In accord with an approach originally outlined by Jortner and coworkers,41 42 the influence of changing AG° upon the 180 KIE has been modeled using a saddle point approximation.43 At this stage, the experimental variations in 180 KIEs for reactions of O2 and O2" are yet to be determined. The vibronic model of Hammes-Schiffer, which has been used to model proton-coupled electron transfer in accord with a Bom-Oppenheimer separation of timescales, may also be applicable here.44 The objective is to account for the change in 0—0 vibrational frequency together with potential contributions from overlap of vibrationally excited states. The overlap factors involving these states are expected to become more important as AG° deviates from 0 kcal mol 1,39... 

In more physical language, the effective rotational hamiltonian in each vibrational state is obtained by averaging the original hamiltonian over the vibrational co-ordinates using the true vibrational wavefunctions, obtained by an appropriate perturbation of the harmonic oscillator basis functions. It is an extension of the Bom-Oppenheimer separation of the electronic from the nuclear motion, to achieve a separation of the vibrational from the rotational motion. 

It is well known from the Bom-Oppenheimer separation [1] that the pattern of energy levels for a typical diatomic molecule consists first of widely separated electronic states (A eiec 20000 cm-1). Each of these states then supports a set of more closely spaced vibrational levels (AEvib 1000 cm-1). Each of these vibrational levels in turn is spanned by closely spaced rotational levels ( A Emt 1 cm-1) and, in the case of open shell molecules, by fine and hyperfine states (A Efs 100 cm-1 and AEhts 0.01 cm-1). The objective is to construct an effective Hamiltonian which is capable of describing the detailed energy levels of the molecule in a single vibrational level of a particular electronic state. It is usual to derive this Hamiltonian in two stages because of the different nature of the electronic and nuclear coordinates. In the first step, which we describe in the present section, we derive a Hamiltonian which acts on all the vibrational states of a single electronic state. The operators thus remain explicitly dependent on the vibrational coordinate R (the intemuclear separation). In the second step, described in section 7.55, we remove the effects of terms in this intermediate Hamiltonian which couple different vibrational levels. The result is an effective Hamiltonian for each vibronic state. 

Having in mind the dramatic effects the establishment of an H-bond has on the I s band-shape, we may anticipate that this anharmonic coupling is not small. It means that it cannot be handled by classical perturbation techniques. It may, however, be taken into account in the frame of the adiabatic separation (6) of rapid and slow motions. This adiabatic separation is already used to separate the motions of the electrons in the molecular complex from the vibrations of the atoms and is then called Bom-Oppenheimer separation. In this approximation applied to the separation of from the intermonomer modes, the rapid vibration I s, which is ruled by H(q,Q ) of eq. (5.2) and displays characteristic wavenumbers around... 

Eq. (5.A37) is equivalent to supposing that the motion in Q is sufficiently slow that its kinetic energy term P cannot induce transitions between various levels n) of the rapid motion. Mathematically it means that all matrix elements of P of the form m P n are zero, unless m = n, in which case it is then equal to P (it still acts on a wavefunction of the slow motion on its right hand). This is the basis of the Bom-Oppenheimer separation between electrons and nuclei in molecules. We may then write, supposing that the electric dipole moment ju(0) displays a linear dependence on normal mode q, an approximation called electrical harmonicity that reveals excellent for most molecular systems... 

Based upon the Bom-Oppenheimer separation of electronic and nuclear energies, the B-0 potential sinface should be isotopically invariant and the stmcture of a molecule should also be isotopically invariant. Most of the ti tly bound molecules of interest to chemists conform very well to the B-0 approximation, although all molecules deviate when examined at sufficient precision. Weakly bound hydrogen-bonded species or van der Waals molecules such as Ar—HCl are exceptions that require special considerations [3]. 

We consider the nature of the interaction between the ion and the wall source for the purpose of determining first the effect of the ion on vibrations of the wall. One properly needs to carry out a Bom-Oppenheimer separation of the wall vibrations from the channel ion motions to be complete. It is probably adequate, however, in view of the approximate character of the model representation, to examine the wall-coordinate dependence of the total ion band energy E(k) alone. As is... 

There is no evidence that any classical attribute of a molecule has quantum-mechanical meaning. The quantum molecule is a partially holistic unit, fully characterized by means of a molecular wave function, that allows a projection of derived properties such as electron density, quanmm potential and quantum torque. There is no operator to define those properties that feature in molecular mechanics. Manual introduction of these classical variables into a quantum system is an unwarranted abstraction that distorts the non-classical picture irretrievably. Operations such as orbital hybridization, LCAO and Bom-Oppenheimer separation of electrons and nuclei break the quantum symmetry to yield a purely classical picture. No amount of computation can repair the damage. 

The potential energy surface for a diatomic molecule can be represented as in figure A 1.2.1. The x -axis gives the intemuclear separation R and the y -axis the potential function V R). At a given value of R, the potential V R) is determined by solving the quantum mechanical electronic stmcture problem in a framework with the nuclei fixed at the given value of R. (To reiterate the discussion above, it is only possible to regard the nuclei as fixed in this calculation because of the Bom-Oppenheimer separability, and it is important to keep in mind that this is only an approximation. 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system waveflmction. In the fixed nuclei approximation, the Bom-Oppenheimer separation of electronic and nuclear motion permits electronic motion and scattering amplitudes f, (R) to be determined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for i = n, V, J n, v, J = /transitions is... 

Methodology relating to a perturbation expansion about an equilibriuni configuration is referred to as the Bom-Oppenheimer approximation and the separation per se of the nuclear and electronic motions is designated as the Bom-Oppenheimer separation. 

In principle, it is well to separate the external constants of the motion (see Section IV-A) before electronic-nuclear separation, as the remaining motion will then correspond more readily with observables. Jepsen and Hirschfelder find, for example, that electronic-nuclear coupling is decreased if translation is removed prior to the Bom-Oppenheimer separation. The same can probably be said for rotational motion, but the difference would be more difficult to compute. 

The electronic hamiltonian includes all the terms of the total hamiltonian but the nuclear kinetic term. Equation (3) is called the electronic Schrodinger equation as it only includes the motion of electrons as variables. The coordinates of the nuclei are only parameters in this equation so that the obtained electronic energy is a function of the nuclear coordinates. This is the potential energy surface (PES). Equation (4) is the nuclear Schrodinger equation. As it contains the PES, it can only be treated once the electronic equation (3) has been solved. This split of the Schrodinger equation in two is known as the Bom-Oppenheimer separation. 

Solving eq. (11) including eq. (12) would aunount to exaud solution of the coupled-chamnel or close-coupling (CC) formulation in the adiabatic representation. The adiabatic approximation consists in neglecting the coupling terms, as it is done in the Bom-Oppenheimer separation. 

The Bom-Oppenheimer separation of the electronic and nuclear motions is a cornerstone in computational chemistry. Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the potential energy surface (PES) is known. The motion of the nuclei on the PES can then be solved either classically (Newton) or by quantum (Schrodinger) methods. If there are N nuclei, the dimensionality of the PES is 3N, i.e. there are 3N nuclear coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. For a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N - 6(5) coordinates to describe the internal movement of the nuclei, which for small displacements may be chosen as vibrational normal coordinates . 

The vibration-rotation spectra and/or the rotational spectra in excited vibrational states provide the af constants and, when all the a/ constants are determined, the equilibrium rotational constants can be obtained by extrapolation. This method has often been hampered by anharmonic or harmonic resonance interactions in excited vibrational states, such as Fermi resonances arising from cubic and higher anharmonic force constants in the vibrational potential, or by Coriolis resonances. Equihbrium rotational constants have so far been determined only for a limited number of simple molecules. To be even more precise, one has further to consider the contributions of electrons to the moments of inertia, and to correct for the small effects of centrifugal distortion which arise from transformation of the original Hamiltonian to eliminate indeterminacy terms [11]. Higher-order time-independent effects such as the breakdown of the Bom-Oppenheimer separation between the electronic and nuclear motions have been discussed so far only for diatomic molecules [12]. 

Some details. Write the (non-relativistic) Hamiltonian as/T= r+//ei, where Tis the kinetic energy operator for the relative motion of the nuclei. For a diatomic molecule, T = —(ft /2/r)V. The electronic Hamiltonian (Hel) is a sum of the kinetic energy operator of the electrons and potential terms, all electrostatic. We list them so as to emphasize that a number of these terms depend on the distance R between the two nuclei the repulsion between the two nuclei, the attraction of the electrons to the two nuclei, and the repulsion between the electrons. In the Bom-Oppenheimer separation we first hold R constant and diagonalize the electronic Hamiltonian = Ee[ R)ir rJi). We... 

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