Fixed-nucleus 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 wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

All of the above-mentioned investigations on the hydrogen atom were performed in the fixed-nuclei approximation, i.e. for the assumption of an infinitely heavy nucleus. In the absence of a magnetic field the center of mass (CM) and electronic motion separate exactly, and the influence of the finite nuclear mass on the behavior and properties of the atom can be taken into... 

One of the most important characteristics of molecular systems is their behavior as a function of the nuclear coordinates. The most important molecular property is total energy of the system which, as a function of the nuclear coordinates, is called the potential energy (hyper)surface, an obvious generalization of the potential energy curve in diatomics. Other expectation values as functions of the nuclear coordinates are frequently called property surfaces. The notion of the total molecular energy in a given electronic state, which depends only parametrically on the nuclear coordinates, is based on the fixed-nuclei approximation. In most cases (e.g. closed-shell molecules in the ground electronic state and in a low vibrational state) this is an excellent approximation. Even when it breaks down, the most convenient treatment is based on the fixed-nuclei picture, i.e. on the assumption that the nuclear mass is infinite compared with the electronic mass. 

The fundamental fixed-nuclei approximation, in which the nuclei are treated as distinguishable classical particles with positions in physical space, allows for the electronic structure of ground and excited states of both atoms and molecules to be ruled by the same principles, concepts, and approximations, such as the occupation of symmetry-adapted atomic or molecular orbitals by electrons, subject to the Pauli exclusion principle. The resulting basic interpretative and computational tool is the N-electron symmetry-adapted configuration (SAC), be it atomic or molecular. It is denoted here by i. When the SAC is adopted as a conceptual and computational tool, it is possible to use the same concepts and theoretical methods in order to treat the electronic eigenfunctions of states of both atoms and small molecules for each fixed geometry. 

Vibronic coupling effects on the chiroptical spectra of optically active metal complexes tend to obscure the inherent relationships between the CD observables and structural features such as absolute configuration, ligand conformation, and ligand spatial distributions. For this reason, spectra-structure relationships based on the "fixed-nuclei" approximation and on the assumption of well defined "electronic" transitions must be applied with considerable caution. A great wealth of spectroscopically and structurally important information may be obtained from detailed vibronic analyses of CD spectra. However, very few analyses of this sort have been reported to date. 

In Chap. 2, we formulate our basic framework of the chemical theory based on the Born-Oppenheimer approximation. We briefly discuss how valid or how accurate the Born-Oppenheimer approximation for bound states is. Also a theory of electron scattering by polyatomic molecules within the Born-Oppenheimer framework (or the so-called fixed nuclei approximation) is presented. This is one of the typical theories of electron dynamics, along with the theory of molecular photoionization. 

The fixed nuclear approximation is extremely useful, not only for the bound states, but also for the treatment of electronic scattering by molecules. For instance, stationary-state scattering theories within the fixed nuclear approximation have been extensively developed for molecular photoionization [18, 250, 336, 516] and electron scattering from polyatomic molecules [210, 418, 419]. These scattering phenomena are quite important in that they are widely foimd in nature as elementary processes and even in industrial applications using plasma processes. These scatterings may be referred to as stationary-state electron dynamics in fixed nuclei approximation. 

Since the publication of the seminal paper by Born and Oppenheimer in 1927, the theoretical foundation of molecular science in the 20th century, and even to date, had been dominated by the so-called Born-Oppenheimer approximation, which dynamically separates electronic and nuclear motions under an assumption that electrons can follow the nuclear dynamics almost instantaneously. This idea led to the fixed nuclei approximation and the notion of electronic stationary-states that adjust themselves to any nuclear configurations in space. 

In 1933 Herzberg and Teller demonstrated that certain electronic transitions which are forbidden in the fixed nuclei approximation may attain a non-zero transition probability through the interaction of electronic and nuclear motions. What they essentially did was to note that in every molecule there exist asymmetric nuclear displacements which yield a non-zero expectation value for the gradient of the nuclear-electronic coulombic energy terms. To illustrate their mode of thought we shall here give a pictorial outline of the calculation of such vibronic intensities in two systems of current interest aromatic compounds and transition metal complexes. For the sake of clarity we shall limit our discussion to the exemplary molecular systems of benzene and potassium titanium hexafluoride (KjTiFg). 

If / vanishes in the fixed nuclei approximation, the electronic jump is said to be electronically forbidden if it vanishes also in the moving nuclei approximation, the spectral transition is said to be vibronically (i.e., vibrationally-electronically) forbidden. Vibronically allowed emissions or absorptions are called Herzberg-Teller bands. 

In order to have a classical picture we have to neglect the nuclear kinetic energy operators (fixed nuclei approximation) ... 

However, in the fixed nuclei approximation, it would not imply that... 

It is important to recognize that this implies nothing else but the fixed-nuclei approximation P (or, for that matter, any molecule) is described as a rigid nuclear framework whose geometry conformation, number and quality of bonds, charge and dipole moment distribution, etc. is entirely determined by the field of the fixed nuclei and the motion of the electrons in it. 

It may be noted that, under the fixed-nuclei approximation, a perturbation consisting in the juxtaposition of a system A to the system P only involves additional terms associated to the kinetic energy of the additional electrons and an additional potential. But if an effective one-electron Hamiltonian is constructed, this reduces to a change in the potential under which the effective electron moves, and the difference in the number of electrons is taken into account in the Hamiltonian if and only if the effective potential is dependent on the density matrix - i.e. if the effective one-electron Hamiltonian is an SCF one. Otherwise, the only consequence of the change in number of electrons will be felt in the occupation scheme, and the Hamiltonian will not be affected by it. 

The n-electron Hamiltonian H in the fixed nucleus approximation may be written as... 

The first is the Bom-Oppenheimer or fixed-nucleus approximation [32] wherein the more massive nuclei are assumed to be stationary with the electrons moving rapidly about them. The solution of Eq (3) reduces to finding the energies and trajectories of the electrons only, i.e. solution of the so-called many-electron Schrodinger equation. A further simplification which is often assumed, at least initially, is that relativistic effects are negligible. The starting point for so-called ab initio methods is therefore the non-relativistic many-electron Schrodinger equation within the Born-Oppenheimer approximation. 

The force on the ath nucleus in this fixed-nucleus approximation is given by... 

Once the potential energy curve E (R ) vs R of a diatomic molecule has been determined from the Schrodinger equation of the electronic problem in the fixed-nucleus approximation (Born-Oppenheimer), there are various methods to determine the force constants and vibrational frequencies non-empirically. These methods will now be described below. 

In Refs. 44. 52. 55) the same potential function used to calculate <01 is also used to calculate the other tetrahedral frequencies o)t, (03 and (04. It is not clear whether V was redetermined for each type of normal mode V(Q ) should be calculated in the fixed nucleus approximation, shifting the atoms along their actual normal-mode trajectories for the various normal modes then a quadratic potential curve should be fitted. This could be explanation for the bad correlation between to calc and to exp which involve motion of the central x-atom and/or bending modes of the H-atoms (see later). 

The Born-Oppenheimer Hamiltonian operator H = (T + V) for the fixed-nucleus approximation (Eq. 2.5) is expanded in powers of the displacement (R — Rg) near Rg and second-order perturbation theory is used to calculate E correct to second-order. Then k is given by... 

To summarize (1.1.1)-(1.1.12) provide a mathematical basis for molecular quantum mechanics in a fixed-nucleus approximation, but the limitations of this theoretical model (which by now should be apparent) must not be forgotten. 

It is interesting to note that the transformation (11.1.15), with A given in (11.1.18), corresponds to a very simple change in the density matrices when we specialize to the case of electronic systems (i.e. the usual fixed-nucleus approximation). For a gauge change in which A— A + grad/, the iV-electron wavefunction must be multiplied by... 

In the following sections we shall use a fixed-nucleus approximation in discussing some of the observable effects of the small terms. That we may do so, in spite of the fact that the presence of velocity-dependent terms precludes the rigorous separation of electronic and nuclear motion (even in the absence of external fields), is due to their dependence on higher powers of u/c—a ratio that is exceedingly small for nuclei. The terms that we shall in fact consider, some of which involve further reduction of the many contributions listed in Appendix 4, are conveniently collected at this point those of most interest may be distinguished by writing H as... 

Here we simply put on record the results obtained (see e.g. Moss, 1973) for a many-electron molecule in the fixed-nucleus approximation. The Hamiltonian will be written (cf. (1.1.13)) in the form... 

According to the classical theory, the effect of a magnetic field on a system composed of electrons in motion about a fixed nucleus is equivalent to the first order of approximation to the imposition on the system of a uniform rotation... 

At the same time that Heisenberg was formulating his approach to the helium system, Born and Oppenheimer indicated how to formulate a quantum mechanical description of molecules that justified approximations already in use in treatment of band spectra. The theory was worked out while Oppenheimer was resident in Gottingen and constituted his doctoral dissertation. Born and Oppenheimer justified why molecules could be regarded as essentially fixed particles insofar as the electronic motion was concerned, and they derived the "potential" energy function for the nuclear motion. This approximation was to become the "clamped-nucleus" approximation among quantum chemists in decades to come.36... 

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... 

For the N-electron atom, we have seen (Section 3.7) several terms in the Hamiltonian operator. We collect here some more terms, to come to a "final list," within the Born-Oppenheimer approximation of a fixed nucleus ... 

The first step in simplifying the SOS-PT formulas is to apply the clamped nucleus approximation for the states K 0 [16]. In this approximation the energy denominators b are replaced by the difference in electronic energies at a fixed nuclear configuration, i.e. by E K, R) - (0, R). The consequences of this approximation have been investigated and were found to be negligible [16]. 

The corresponding wavefunction must then include both electronic and nuclear coordinates. The validity of the fixed-nucleus model discussed so far was first established by Bom and Oppenheimer (1927) (see also Born and Huang, 1954), who expanded the total molecular wavefunction in terms of products of electronic and nuclear wavefunctions, and showed that in good approximation a single product was usually appropriate the electronic wavefunction is then a solution of (1.1.1), while the nuclear wavefunction is derived from a nuclear eigenvalue equation in which obtained in (1.1.10), as a function of nuclear positions (via solution of (1.1.1)), is used as a potential function. It is because of the validity of this separation, which depends on the large ratio between electronic and nuclear masses, that we may confine our attention initially to a purely electronic problem. 

To overcome the primary weakness of GTO fimetions (i.e. their radial derivatives vanish at the nucleus whereas the derivatives of STOs are non-zero), it is coimnon to combine two, tliree, or more GTOs, with combination coefficients which are fixed and not treated as LCAO-MO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs are multiplied by contraction coefficients and suimned to produce a CGTO, which approximates the proper cusp at the nuclear centre. 

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

The separation of the electronic and nuclear motions depends on the large difference between the mass of an electron and the mass of a nucleus. As the nuclei are much heavier, by a factor of at least 1800, they move much more slowly. Thus, to a good approximation the movement of the elections in a polyatomic molecule can be assumed to take place in the environment of the nuclei that are fixed in a particular configuration. This argument is the physical basis of the Born-Oppenheimer approximation. 

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

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