Solute electronic wavefunction

The effective electronic Hamiltonian, /7eff, for the solute has already been introduced in the contribution by Tomasi. It describes the solute under the effect of the interactions with its environment and determines how these interactions affect the solute electronic wavefunction and properties. The corresponding effective Schrodinger equation reads... 

In both cases we can introduce a similar picture in terms of an effective Hamiltonian giving rise to an effective Schrodinger equation for the solvated solute. Introducing the standard Born-Oppenheimer approximation, the solute electronic wavefunction ) will satisfy the following equation ... 

For charge transfer reactions, the solute electronic wavefunction is expanded in a basis of VB (also denoted diabatic ) charge transfer states... 

Vint may be modeled in many different ways. One of the extreme examples is the solvation model proposed years ago by Klopman (1967), which is quoted here to show the flexibility of this approach, and not to suggest its use (the limits of this model have been known since a long time). In this model each nucleus of M is provided with an extra phantom charge (the solvaton), which introduces, via Coulombic interactions, a modification of the solute electronic wavefunction and of the expectation value of the energy, Em, mimicking solvent effects. 

This form is particularly useful if the solute system can undergo a chemical reaction. Both factors in the RF depend on the global nuclear configuration X. The nuclear configuration(s) can be obtained either by sampling the space with MD or MC procedures, or it can be obtained, for particular systems, from X-ray and/or neutron diffraction methods. In crystals and protein surroundings an average structure can be defined [23, 26, 27]. The fluctuations around such structures can be sampled with MC or MD procedures. In this manner, structural fluctuation effects on the solute electronic wavefunction, and associated properties can be subjected to numerical calculations. 

The electronic wavefunction is thus given as solution of = ggiAe and the total energy is given by... 

Solutions to the Schrodinger equation (3.5) are called one-electron wavefunctions or orbitals and take the form in Eq. (3.6)... 

Ab initio calculations usually begin with a solution of the Hartree-Fock equations, which assumes the electronic wavefunction can be written as a single determinant of molecular orbitals. The orbitals are described in terms of a basis set of atomic functions and the reliability of the calculation depends on the quality of the basis set being used. Basis sets have been developed over the years to produce reliable results with a minimum of computational cost. For example, double zeta valence basis sets such as 3-21G [15] 4-31G [16] and 6-31G [17] describe each atom in the molecule with a single core Is function and two functions for the valence s and p functions. Such basis sets are commonly used, as there appears to be a cancellation of errors, which fortuitously allows them to predict quite accurate results. 

The simplest solution to this problem is to construct an antisymmetric wavefunction using a linear combination of one-electron wavefunctions. For two electrons, this takes the following form ... 

This integral is that of a core-electron interaction and therefore available through solution of the many-electron wavefunction using a variety of methods. 

For all but the simplest systems the Schrodinger equation must be solved approximately. It is assumed that the true wavefunction, which is too complicated to be found directly, can be approximated by a simpler function. For some types of function it is then possible to solve the electronic Schrodinger equation numerically. Provided the assumption made regarding the form of the function is not too drastic, a good approximation will be obtained to the correct solution. Electronic structure theory consists of designing sensible approximations to the wavefunction, with an inevitable trade-off between accuracy and computational cost. 

The most usual starting point for approximate solutions to the electronic Schrodinger equation is to make the orbital approximation. In Hartree-Fock (HF) theory the many-electron wavefunction is taken to be the antisymmetrized product of one-electron wavefunctions (spin-orbitals) ... 

It is known (Chap. A) that Koopmans theorem is not vahd for the wavefunctions and eigenvalues of strongly bound states in an atom or in the cores of a solid, i.e. for those states which are a solution of the Schrodinger (or Dirac) equation in a central potential. In them the ejection (or the emission) of one-electron in the electron system means a strong change in Coulomb and exchange interactions, with the consequent modification of the energy scheme as well as of the electronic wavefunction, in contradiction to Koopmans theorem. 

Having decided to use AOs (or combinations of them) for yrA and pB> we will now look at the form these take. They are approximate solutions to the Schrodinger equation for the atom in question. The Schrodinger equation for many-electron atoms is usually solved approximately by writing the total electronic wavefunction as the product of one-electron functions (these are the AOs). Each AO 4>i is a function of the polar coordinates r, 0, and single electron and can be written as... 

In the usual way (74,75), one introduces a complete orthogonal set of electronic wavefunctions which are solutions of... 

The use of this expression for a variational determination of T is a complex problem because of the /V-representability requirement [15, 16, 17], Nevertheless, there is a renewed interest in this problem and a number of methods, including so called cumulant-based approximations [18, 19] are being put forth as solutions to the representability problem. Although some advances can be obtained for special cases there appears to be no systematic scheme of approximating the density matrix with a well-defined measure of the N-representability error. Obviously, the variational determination of density matrices that are not guaranteed to correspond to an antisymmetric electronic wavefunction can lead to non-physical results. 

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