Born-Oppenheimer product function

If these Born-Oppenheimer product wave functions are to approximate Hamiltonian eigenvectors, we have to minimize all off-diagonal matrix elements K L and u v). To this end, the electronic wave functions are chosen to be eigenvectors of a part of the Hamiltonian operator called the electronic Hamiltonian (adiabatic states) ... 

Within the Born-Oppenheimer (BO) approximation, A) and B) may be written as the product of an electronic wave function, M)el and a nuclear wave function M)n. 

The problem, as Woolley addressed it, is that quantum mechanical calculations employ the fixed, or "clamped," nucleus approximation (the Born-Oppenheimer approximation) in which nuclei are treated as classical particles confined to "equilibrium" positions. Woolley claims that a quantum mechanical calculation carried out completely from first principles, without such an approximation, yields no recognizable molecular structure and that the maintenance of "molecular structure" must therefore be a product not of an isolated molecule but of the action of the molecule functioning over time in its environment.47... 

In principal one can calculate the electronic energy as a function of the Cartesian coordinates of the three atomic nuclei of the ground state of this system using the methods of quantum mechanics (see Chapter 2). (In subsequent discussion, the terms coordinates of nuclei and coordinates of atoms will be used interchangeably.) By analogy with the discussion in Chapter 2, this function, within the Born-Oppenheimer approximation, is not only the potential energy surface on which the reactant and product molecules rotate and vibrate, but is also the potential... 

Fig. 1. The Marcus parabolic free energy surfaces corresponding to the reactant electronic state of the system (DA) and to the product electronic state of the system (D A ) cross (become resonant) at the transition state. The curves which cross are computed with zero electronic tunneling interaction and are known as the diabatic curves, and include the Born-Oppenheimer potential energy of the molecular system plus the environmental polarization free energy as a function of the reaction coordinate. Due to the finite electronic coupling between the reactant and charge separated states, a fraction k l of the molecular systems passing through the transition state region will cross over onto the product surface this electronically controlled fraction k l thus enters directly as a factor into the electron transfer rate constant...

Since the exact solution of Schrodinger s equation for multi-electron, multi-nucleus systems turned out to be impossible, efforts have been directed towards the determination of approximate solutions. Most modern approaches rely on the implementation of the Born-Oppenheimer (BO) approximation, which is based on the large difference in the masses of the electrons and the nuclei. Under the BO approximation, the total wave-function can be expressed as the product of the electronic il/) and nuclear (tj) wavefunctions, leading to the following electronic and nuclear Schrodinger s equations ... 

Before we can discuss the recent developments further, we must discuss terminology. This will also provide a guide to earlier literature as well as to the classification we use in our review of recent papers (Section lOd). As mentioned, the total wave function in the Born-Oppenheimer method. is expressed as a product of electronic and vibrational wave functions. What has resulted is that different types of electronic wave functions have been used (which is not necessarily confusing), and that in many cases a particular selection has been called the Born-Oppenheimer approximation (which has led to confusion). We discuss here only the two predominant choices of... 

To deduce whether a transition is allowed between two stationary states, we investigate the matrix element of the electric dipole-moment operator between those states (Section 3.2). We will use the Born-Oppenheimer approximation of writing the stationary-state molecular wave functions as products of electronic and nuclear wave functions ... 

The Born-Oppenheimer approximation separates the molecular wave function into a product of electronic and nuclear wave functions ... 

In the Born-Oppenheimer approximation, the molecular wave function is the product of electronic and nuclear wave functions see (4.90). We now examine the behavior of if with respect to inversion. We must, however, exercise some care. In finding the nuclear wave functions fa we have used a set of axes fixed in space (except for translation with the molecule). However, in dealing with if el (Sections 1.19 and 1.20) we defined the electronic coordinates with respect to a set of axes fixed in the molecule, with the z axis being the internuclear axis. To find the effect on if of inversion of all nuclear and electronic coordinates, we must use the set of space-fixed axes for both fa and if el. We shall call the space-fixed axes X, Y, and Z, and the molecule-fixed axes x, y, and z. The nuclear wave function of a diatomic molecule has the (approximate) form (4.28) for 2 electronic states, where q=R-Re, and where the angles are defined with respect to space-fixed axes. When we replace each nuclear coordinate in fa by its negative, the internuclear distance R is unaffected, so that the vibrational wave function has even parity. The parity of the spherical harmonic Yj1 is even or odd according to whether J is even or odd (Section 1.17). Thus the parity eigenvalue of fa is (- Yf. 

We have used the Born Oppenheimer approximation to factor 4 0/3, I,ma into electronic and nuclear parts and have further assumed that the former are orthogonal to enable us to reduce V. Both wave functions may be approximated by products of electronic, nuclear rotation and vibrational wave functions. The last of these may be factored out at once, and... 

In addition, we assume, for the systems of interest here, that the electronic motion is fast relative to the kinetic motion of the nuclei and that the total wave functions can be separated into a product form, with one term depending on the electronic motion and parametric in the nuclear coordinates and a second term describing the nuclear motion in terms of adiabatic potential hypersurfaces. This separation, based on the relative mass and velocity of an electron as compared with the nucleus mass and velocity, is known as the Born-Oppenheimer approximation. 

In the Born-Oppenheimer approximation, the /ith well state (r, Q) for the well centered at the point Qn is given by the product of the electronic wave function wave function o(2 Qn) of the nuclear vibrations so that... 

The theory of multi-oscillator electron transitions developed in the works [1, 2, 5-7] is based on the Born-Oppenheimer s adiabatic approach where the electron and nuclear variables are divided. Therefore, the matrix element describing the transition is a product of the electron and oscillator matrix elements. The oscillator matrix element depends only on overlapping of the initial and final vibration wave functions and does not depend on the electron transition type. The basic assumptions of the adiabatic approach and the approximate oscillator terms of the nuclear subsystem are considered in the following section. Then, in the subsequent sections, it will be shown that many vibrations take part in the transition due to relative change of the vibration system in the initial and final states. This change is defined by the following factors the displacement of the equilibrium positions in the... 

It is not difficult to see the asymptotics of the wave functions (7, q E,) ri i() (56) that are not represented, generally speaking, by the product of the wave functions of the electron and nuclear coordinates, contrary to the wave function in MREL (48). The contributions of the different vibration modes in the asymptotics (56) change with changes in the electron coordinates. So, Born-Oppenheimer s approach is really violated in the asymptotics of the wave function [2], Such violation has also been revealed in article [22]. 

In the Born-Oppenheimer approximation the basis set for 3Q,i would consist of products of electronic space and spin functions. Transformation to the gyrating axis system may involve transformation of both space and spin variables, leading to a Hamiltonian in which the spin is quantised in the molecule-fixed axis system (as, for example, in a Hund s case (a) coupling scheme) or transformation of spatial variables only, in which case spatially quantised spin is implied (for example, Hund s case (b)). We will deal in detail with the former transformation and subsequently summarise the results appropriate to spatially quantised spin. 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, while their kinetic energy includes all kinds of motion, vibration and rotation as well as translation. First, as in the case of atoms, the translational motion of the molecule is isolated. Then a two-step approximation can be introduced. The first is the separation of the rotation of the molecule as a whole, and thus the remaining equation describes only the internal motion of the system. The second step is the application of the Born-Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei move much more slowly than the electrons, the latter can be assumed to move about a fixed nuclear arrangement. Accordingly, not only the translation and rotation of the whole molecular system but also the internal motion of the nuclei is ignored. The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of both nuclei and electrons but it is solved for the motion of the electrons only. 

Separation of Electronic and Nuclear Motion. Because, in general, electrons move with much greater velocities than nuclei, to a first approximation electron and nuclear motions can be separated (Born-Oppenheimer theorem [3]). The validity of this separation of electronic and nuclear motions provides the only real justification for the idea of a potential-energy curve of a molecule. The eigenfunction Y for the entire system of nuclei and electrons can be expressed as a product of two functions F< and T , where is an eigenfunction of the electronic coordinates found by solving Schrodinger s equation with the assumption that the nuclei are held fixed in space and Yn involves only the coordinates of the nuclei [4]. 

When discussing symmetry selection rules it was mentioned that vibrational motion can influence both the shape and the intensity of electronic absorption bands. In the usual Born-Oppenheimer approximation with molecular wave functions written as products as in Equation (1.12) this can be understood as follows. 

Here im is the effective mass of the i th vibration and Pi is the momentum conjugate to the corresponding normal vibrational coordinate Qi. The first two terms transform the electronic levels into potential energy manifolds in the coordinates of the octahedral normal modes Qi with vibrational frequencies m,- = yZ T/I/", and the complete wave functions in the Born-Oppenheimer approximation can be written as a product of the electronic and vibrational parts. The third term describes the distortions produced by the vibrations and can be interpreted in terms of a force Fi, which acts along the vibrational mode Qi associated with the electronic state E ... 

The Born-Oppenheimer approximation allows us to decouple the electronic and nuclear motions of the free molecule of the Hamiltonian Hq. Solving the Schrodinger equation //o l = with respect to the electron coordinates r = r[, O, gives rise to the electronic states (r, R) = (r n(R)), n = 0,..., Ne, of respective energies En (R) as functions of the nuclear coordinates R, with the electronic scalar product defined as (n(R) n (R))r = j dr rj( r. R) T,-(r, R). We assume Ne bound electronic states. The Floquet Hamiltonian of the molecule perturbed by a field (of frequency co, of amplitude 8, and of linear polarization e), in the dipole coupling approximation, and in a coordinate system of origin at the center of mass of the molecule can be written as... 

H. We understood H to be complete and including electronic as well as nuclear degrees of freedom, and in which case the states are the true nonadiabatlc vlbronic eigenstates of the system and hence the properties are the exact ones. Nothing prevents us, however, to introduce the adiabatic approximation and to assume the wave functions to be products of electronic and nuclear (vibrational) parts. In this case, the Born-Oppenheimer electronic plus vibrational properties will appear. We can even reduce the accuracy to the extent that we adopt the electronic Hamiltonian, work with the spectrum of electronic states, and thus extract the electronic part of the properties. In all these cases, the SOS property expressions remain unchanged. 

Whereas alteration of enzyme or substrate structure causes large perturbations to the potential energy functions describing enzyme-substrate interactions, a consequence of the Born-Oppenheimer approximation is that isotopic substitution causes no perturbation of potential energy functions at all. Isotope effects are therefore amongst the most powerful methods of determining enzyme mechanism. If they are measured by any sort of competition method (for example, isotope enrichment in the unreacted starting material or product) then, because one isotopomer acts as a competitive inhibitor of the... 

It might also be hoped that a first approximation to a solution of (16) could be constructed in terms of product functions in which one portion of each product was obtained from a problem in which the electronic motion was treated as primary, and the other portion described the nuclear motion in the electronic field derived from the first part of the product. This is the standard technique for treating a system of coupled differential equations in which one group of equations represent fast motions and another slow motions. Most chemists get their familiarity with this technique by considering the kinetics of sequential chemical reactions. It is this technique that underlies the Born and Oppenheimer program in which the electronic motion is approached in a frame fixed in the laboratory with an electronic Hamiltonian in which the nuclear motion is at first ignored. Thus, it is natural with the present coordinate choice to hope that functions of the form... 

The potential-energy term couples nuclear and electronic coordinates and cannot be neglected. Therefore, electronic and nuclear motion should be coupled. However, because nuclei are much heavier than electrons, the Born-Oppenheimer approximations is generally valid. In the Born-Oppenheimer approximation, the molecular wave function is written as a sum of products of electronic wave functions Fj,(q Q), explicitly dependent on electronic coordinates and parametrically dependent on nuclear coordinates, and nuclear wave functions... 

The full multiple spawning (FMS) method has been developed as a genuine quantum mechanical method based on semiclassical considerations. The FMS method can be seen as an extension of semiclassical methods that brings back quantum character to the nuclear motion. Indeed, the nuclear wave function is not reduced to a product of delta functions centered on the nuclear positions but retains a minimum uncertainty relationship. The nuclear wave function is expressed as a sum of Born-Oppenheimer states ... 

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

Note that the wave functions for the initial and final states and include both donor and acceptor. This equation is usually simplified by making the Born-Oppenheimer approximation for the separation of nuclear and electron wave functions, resulting in equation (12), in which V is the electronic matrix element describing the coupling between the electronic state of the reactants with those of the product, and FC is the Franck-Condon factor. 

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