Independent-particle model, wave function

The various methods used in quantum chemistry make it possible to compute equilibrium intermolecular distances, to describe intermolecular forces and chemical reactions too. The usual way to calculate these properties is based on the independent particle model this is the Hartree-Fock method. The expansion of one-electron wave-functions (molecular orbitals) in practice requires technical work on computers. It was believed for years and years that ab initio computations will become a routine task even for large molecules. In spite of the enormous increase and development in computer technique, however, this expectation has not been fulfilled. The treatment of large, extended molecular systems still needs special theoretical background. In other words, some approximations should be used in the methods which describe the properties of molecules of large size and/or interacting systems. The further approximations are to be chosen carefully this caution is especially important when going beyond the HF level. The inclusion of the electron correlation in the calculations in a convenient way is still one of the most significant tasks of quantum chemistry. 

The major difficulty in wave function based calculations is that, starting from an independent-particle model, correlation between electrons of opposite spin must somehow be introduced into T. Inclusion of this type of electron correlation is essential if energies are to be computed with any degree of accuracy. How, through the use of multiconfigurational wave functions, correlation between electrons of opposite spin is incorporated into is the subject of Section 3.2.3. 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

A very important conceptual step within the MO framework was achieved by the introduction of the independent particle model (IPM), which reduces the AT-electron problem effectively to a one-electron problem, though a highly nonlinear one. The variation principle based IPM leads to Hartree-Fock (HF) equations [4, 5] (cf. also [6, 7]) that are solved iteratively by generating a suitable self-consistent field (SCF). The numerical solution of these equations for the one-center atomic problems became a reality in the fifties, primarily owing to the earlier efforts by Hartree and Hartree [8]. The fact that this approximation yields well over 99% of the total energy led to the general belief that SCF wave functions are sufficiently accurate for the computation of interesting properties of most chemical systems. However, once the SCF solutions became available for molecular systems, this hope was shattered. 

In order to define orbitals in a many-electron system, two approaches are possible, which we may refer to as constructive and analytic . The first approach is more common one makes the ad hoc postulate that every electron can be associated with one orbital and the total wave function can be constructed from these orbitals. Then, one is led to an effective one-electron Schrodinger equation from one electron in the field of the other electrons. The underlying model is the independent particle model (IPM). When following the constructive way, one does not know a priori whether the model is a good approximation to the actual physical situation one only knows that it cannot be rigorously correct. The merit of this approach is its relative simplicity from both the mathematical and physical points of view. 

If the wave function of a w-electron system is constructed from individual orbitals in the sense of the independent-particle model, y will have the form. 

In the frame of the independent particle model, the total wave function can be written as an antisymmetrized product of spin orbitals ... 

If a -electron wave function is limited to a Slater determinant of n spin orbitals, one stays within the frame of the independent-particle model, and the best model of that sort (for a discussion, see 22>) for a given problem is that in which the orbitals used to construct the wave function are solutions of the Hartree-Fock equations. This model is only an approximation of the correct wave function. As mentioned in Sect. 3.1, the wave function should be written as a linear combination of Slater determinants, as in Eq. (3.4). To illustrate this, let us consider a two-electron system where the spin can be separated off, so that it is sufficient to consider a function ip (1,2) depending only on the space coordinates of the two particles 1 and 2. For a singlet state ip (1,2) is symmetric with respect to space coordinates ... 

The definition and properties of B(r) may be summarized as follows. For simplicity, we treat the spinless one-electron wave function assuming the independent-particle model or the natural orbital expansion (Lowdin, 1955 Benesch and Smith, 1971). Based on the three-dimensional momentum density p(p),... 

To set up the total time-independent wave function of the many-electron system the independent-particle model is used in general, resulting in an antisymmetrized Hartree product of four-component orthonormal one-electron functions. Independent of the system (atom, molecule or solid) these four-component single electron functions, the spinors, may be written as... 

Is it really necessary to go through the calculations associated with the independent-particle model (which is the basis for the introduction of the correlation concept) as an essential step towards highly accurate wave functions ... 

To analyze correlation effects, it is convenient to refer to a standard independent-particle model. Remembering that the occupation numbers tii of the exact natural orbitals are either close to 2 and sometimes 1 (strongly occupied orbitals and unpaired-electron orbitals) or close to 0 (weakly occupied orbitals), the following process seems to be the simplest way construct an IPM from a set of MO s whose occupation numbers are exactly ( = 2 or 1, i.e. construct an approximate wave function from doubly and possibly singly occupied MO s and supplement it by a set of virtual orbitals to be used for studying the effect of correlation by Cl. 

For an antis5nnmetrized independent-particle model represented by a single-determinant wave function, only the correlation function / (1,2) is identically equal to zero and one has... 

An alternative way of visualizing multi-variable functions is to condense or contract some of the variables. An electronic wave function, for example, is a multi-variable function, depending on 3N electron coordinates. For an independent-particle model, such as Hartree-Fock or density functional theory, the total (determinantal) wave function is built from N orbitals, each depending on three coordinates. 

In fact, the Hartree approximation is the independent-particle model. However, every quantum state in the Hartree wave function satisfies the Pauli exclusion principle such as exchanging the i-th and y-th electrons... 

The geminal ansatz still requires more effort than the standard one-electron approach of the independent particle model. It is therefore usually restricted to small molecules for feasibility reasons. As an example how the nonlinear optimization problem can be handled we refer to the stochastic variational approach [340]. However, the geminal ansatz as presented above has the useful feature that all elementary particles can be treated on the same footing. This means that we can actually use such an ansatz for total wave functions without employing the Born-Oppenheimer approximation, which exploits the fact that nuclei are much heavier than electrons. Hence, electrons and nuclei can be treated on the same footing [340-342] and even mixed approaches are possible, where protons and electrons are treated in the external field of heavier nuclei [343-346]. The integrals required for the matrix elements are hardly more complicated than those over one-electron Gaussians [338,339,347]. 

Because the one-electron operators are identical in form to the one-electron operator in hydrogen-like systems, we use for the independent particle model of Eq. (8.96) for the basis of the many-electron wave function a product consisting of N such hydrogen-like spinors. This ansatz allows us to treat the nonradial part analytically. The radial functions remain unknown. In principle, they may be expanded into a set of known basis functions, but we focus in this chapter on numerical methods, which can be conveniently employed for the one-dimensional radial problem that arises after integration of all angular and spin degrees of freedom. 

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