Wave independent particle methods

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 variational method of quantum chemistry for the determination of the energy has a direct analog in QMC. This is a consequence of the capability of the MC method to perform integration, and should not be confused with MC integration of the integrals that arise in basis set expansion methods. An important branch of QMC is the development of compact and accurate wave functions characterized by explicit dependence on interparticle distances electron-electron and electron-nucleus that are typically written as a product of an independent particle function and a correlation function. Such wave functions lead one immediately to the VMC method for evaluation [3-5]. Wave functions constructed following VMC can also serve as importance functions for the more accurate DMC variant of QMC. 

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

The coupled-cluster method (CCM) is based on the ansatz that an exact many-particle wave function can be written as an exponential cluster operator acting on an independent-particle function,... 

In the LDA, Adolph and Bechstedt [157,158] adopted the approach of Aspnes [116] with a plane-wave-pseudopotential method to determine the dynamic x of the usual IB V semiconductors as well as of SiC polytypes. They emphasized (i) the difficulty to obtain converged Brillouin zone integration and (ii) the relatively good quality of the scissors operator for including quasiparticle effects (from a comparison with the GW approximation, which takes into account wave-vector- and band-dependent shifts). Another implementation of the SOS x —2 ffi, ffi) expressions at the independent-particle level was carried out by Raskheev et al. [159] by using the linearized muffin-tin orbital (LMTO) method in the atomic sphere approximation. They considered... 

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. 

MOs first appear in the framework of the Hartree-Fock (HF) method, which is a mean-field treatment [17,22]. The basic idea is to start from an A-particle wave-function that is appropriate for a system of non-interacting electrons. Having fixed the Ansatz for the A-particle wavefunetion in this way, the variational principle is used in order to obtain the best possible approximation for the fully interacting system. Such independent particle wavefunctions are Slater-determinants, which consist of antisymmetrized products of single-particle wavefunctions (x)J (the antisymmetry brought about by the determinantal form is essential in order to satisfy die Pauh principle). Thus, the Slater-determinant is written as... 

In the above discussion we have been concerned with the exact electronic Hamiltonian, energies and wave functions of a supersystem consisting of an array of well-separated subsystems. We now turn our attention to the description afforded by some independent particle model, in which the electrons move in some mean field. The most commonly used approximation of this type is the Hartree-Fock model, but the discussion presented in this section is not restricted to this particular method. In particular, we write the total electronic Hamiltonian operator in the form... 

Various difficulties of classical physics, including inadequate description of atoms and molecules, led to new ways of visualizing physical realities, ways which are embodied in the methods of quantum mechanics. Quantum mechanics is based on the description of particle motion by a wave function, satisfying the Schrodinger equation, which in its time-independent form is ... 

In the independent-partlcle-model (IPM) originally due to Bohr [1], each particle moves under the Influence of the outer potential and the average potential of all the other particles in the system. In modem quantum theory, this model was first Implemented by Hartree 12], who solved the corresponding one-electron SchrSdlnger equation by means of an iterative numerical procedure, which was continued until there was no change in the slgniflcant figures associated with the electric fields involved so that these could be considered as self-consistent this approach was hence labelled the Self-Conslstent-Fleld (SCF) method. In order to take the Pauli exclusion principle into account. Waller and Hartree [3] approximated the total wave function for a N-electron system as a product of two determinants associated with the electrons of... 

Electron correlation is the phenomenon of the motion of pairs of electrons in atoms or molecules being connected ( correlated ) [56]. The purpose of post-HF calculations is to treat such correlated motion better than does the HF method. In the HF treatment, electron-electron repulsion is handled by having each electron move in a smeared-out, average electrostatic field due to all the other electrons (sections 5.2.3.2 and 5.2.3.6b), and the probability that an electron will have a particular set of spatial coordinates at some moment is independent of the coordinates of the other electrons at that moment. In reality, however, each electron at any moment moves under the influence of the repulsion, not of an average electron cloud, but rather of individual electrons (in fact current physics regards electrons as point particles - with wave properties of course). The consequence of this is that the motion of an electron in a real atom or molecule is more complicated than that for an electron moving in a smeared-out field [57] and the electrons are thus better able to avoid one another. Because of this enhanced (compared to the HF treatment) standoffishness, electron-electron repulsion is really smaller than... 

Originally, the Brueckner theory was derived in a very different way for infinite nuclear matter where the orbitals are plane waves. The Hartree-Fock method cannot be taken as a starting point for nuclear matter, because there the matrix elements over the hard-core nucleon-nucleon repulsions are infinite. These interactions are so strong that it cannot be simply assumed that particles move independently in an undisturbed sea . Brueck-ner s theory essentially cancelled these infinities by including a correlation potential in the potential of the sea . This was... 

So far, we focused on conventional quantum chemical approaches that approximate the FCI wave function by truncating the complete N-particle Hilbert space based on predefined configuration selection procedures. In a different approach, the number of independent Cf coefficients can be reduced without pruning the FCI space. This is equivalent to seeking a more efficient parameterization of the wave function expansion, where the Cl coeflBcients are approximated by a smaller set of variational parameters that allow for an optimal representation of the quantum state of interest. Different approaches, which we will call modern solely to distinguish them from the standard quantum chemical methods, have emerged from solid-state physics. 

Hartree-Fock method A method of solving the Schrodinger equation that assumes that particles motions are independent of each other and a given particle interacts only with the averaged charge distribution of other particles. The electronic wave function can then be approximated by an antisymmetrized product of one-electron functions (orbitals). The effects neglected by the Hartree-Fock method are called correlation effects. 

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