Wave functions, factoring into electronic

In discussing the parity of the molecular wave functions, we have assumed that ip can be factored into electronic, vibrational, and rotational factors ... 

Natural orbitals were defined by P. O. Lbwdin as the eigenfunctions of the first-order reduced density matrix. They provide a simple factorization of wave functions for two-electron systems which brings them into a standard, easily interpreted form. For many-electron systems, they provide a basis for constructing Slater determinants so that the importance of the first few terms is maximized. Used in this way, they were the basis for the construction of some of the first accurate wave functions for molecules. With modern computers, the number of Slater determinants involved in the wave function is no longer so much of an issue and natural orbitals are now mainly used to reduce the wave function and density matrix to a reasonably compact form that facilitates interpretation. 

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

If the spin-orbit coupling is small, as it is in helium, the total electronic wave function J/ can be factorized into an orbital part J/° and a spin part pl ... 

A characteristic feature of the two-electron problem is that the total wave function may be factorized into a space part and a spin part ... 

The study of the two-electron systems was greatly simplified by the fact that the total wave function could be factorized into a space part and a spin part according to Eq. III. I. ForiV = 3, 4,. . , such a separation of space and spin is no longer possible, and an explicit treatment of the spin is actually needed in considering correlation effects. This question of the connection between space and spin in an antisymmetric spin function is a rather complicated problem, which has been brought to a simple solution first during the last few years. 

We note that it is possible to combine the method with correlation factor with the method using superposition of configurations to obtain any accuracy desired by means of comparatively simple wave functions. For a very general class of functions g(r12), one can develop the quotient (r r2)lg(r12) according to Eq. III.2 into products of one-electron functions y>k(r), which leads to the expansion... 

Following the discussion in connection with the expansion III. 127, we note that, for a molecular or a solid-state system, the wave function III. 129 will lead to a correct asymptotic behavior of the energy for separated atoms, provided that the factor g has been conveniently chosen so that it increases indefinitely when any one of the electrons is taken away from the others. A more detailed study of g may sometimes be necessary in order to ensure that no excessive accumulation of ions will occur when the system is separated into its constituents. 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

For three or more electrons, the wave function cannot be factored into a simple product of a space part and a spin part see the next section. 

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... 

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

In Fig. 4.8 the effect of the initial-state wave functions is explored, for the case where the crucial electron-electron interaction is the two-body Coulomb interaction (4.14a) and for the case where this interaction is the two-body contact interaction (4.14d), which is not restricted to the position of the ion. In both cases, the form factor includes the function (4.23), which favors momenta such that pi + p2 is large. This is clearly visible for the contact interaction (4.14d) and less so for the Coulomb interaction (4.14a) whose form factor also includes the factor (4.19), which favors pi = 0 (or p2 = 0)- We conclude that (i) the effect of the specific bound state of the second electron is marginal and (ii) that a pure two-body interaction, be it of Coulomb type as in (4.14a) or contact type as in (4.14d), yields a rather poor description of the data. A three-body effective interaction, which only acts if the second electron is positioned at the ion, provides superior results, notably the three-body contact interaction (4.14b), cf. the left-hand panel (d). This points to the significance of the interaction of the electrons with the ion, which so far has not been incorporated into the S-matrix theory beyond the very approximate description via effective three-body interactions such as (4.14b) or (4.14c). 

Another criticism of the usual MO wave functions is that even at 7 o they overemphasize ionic terms since electron-electron interactions (electron correlations ) are not adequately introduced. According to the simple MO theory, the chance of a given electron being on. a given atom is independent of whether another electron of opposite spin is already there. This cannot be true, and the problem of introducing into the formalism adequate electron correlations to account for this fact has proven a formidable obstacle for the theory of molecules and solids (400). If X = cn/cn = c2i/c22, the space part of equation 24, with normalization factor neglected, can be written in the form... 

In this chapter we show how the separation of the quantum mechanical problem into translational, rotational, vibrational and electronic parts can be achieved. The basis of our approach is to define coordinates which describe the various motions and then attempt to express the wave function as a product of factors, each of which depends only on a small sub-set of coordinates, along the lines ... 

---