Wave function basic principles

We note that such a wave function also fulfills the basic symmetry law derived from the identity principle. 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

From the point of view of principles, it is interesting to note that the method based on the generalized form of Eq. III. 129 seems to be very closely connected both with Wigner s classical theory described in Section III.B and with Bohm and Pines plasma model (Krisement 1957). Following Krisement, we will replace the various trial functions flt /2,. . ., fn in Eq. III.9 by a single average function /, and Wigner s basic wave function (Eq. II1.7) takes then the simple form... 

In the HF scheme, the first origin of the correlation between electrons of antiparallel spins comes from the restriction that they are forced to occupy the same orbital (RHF scheme) and thus some of the same location in space. A simple way of taking into account the basic effects of the electronic correlation is to release the constraint of double occupation (UHF scheme = Unrestricted HF) and so use Different Orbitals for Different Spins (DODS scheme which is the European way of calling UHF). In this methodology, electrons with antiparallel spins are not found to doubly occupy the same orbital so that, in principle, they are not forced to coexist in the same spatial region as is the case in usual RHF wave functions. 

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

A basic principle in quantum mechanics is the indistinguishability of particles. Thus, as indicated in Section 10.5, two particles of the same type in an ideal gas are characterized by a wavefunction, say f(r, 0j, tp 0%, spherical polar coordinates. If for simplicity this wave-function is written as (1,2), the permutation of the coordinates of the two identical particles can be represented by... 

According to the basic principle of quantum mechanics, any measurable property can be computed ab initio if the total wave function y/ describing the quantum eigenstate of the system is known, since it contains the complete... 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

It has been shown that die BCS theory does lead to die phenomenological equations of London. Pippard and Ginzburg and Landau, and one may therefore state that the basic phenomena of superconductivity are now understood from a microscopic point of view, i.e., in terms of the atomic and electronic structure of solids. It is true, however, that we cannot yet, ub initio, calculate V For a given metal and therefore predict whether it will be superconducting or not. The difficulty here is our ignorance of the exact wave functions to be used in describing the electrons and phonons in a specific metal, and their interactions. However, we believe that the problem is soluble in principle at least. 

Arriving subsequently at rigorous quantum mechanical descriptions, we have assumed that the reader has some preliminary knowledge of basic quantum mechanical formalism. We consider it methodologically important to illustrate the correspondence principle between quantum and classical concepts, in particular between the concept of coherence of the wave functions of magnetic sublevels, and the symmetry properties of spatial angular momenta distribution. 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

The basic principle of VB theory is that a covalent bond forms when orbitals of two atoms overlap and the overlap region, which is between the nuclei, is occupied by a pair of electrons. ( Orbital overlap is another way of saying that the two wave functions are in phase, so the amplitude increases between the nuclei.) The central themes of VB theory derive from this principle ... 

The exact meaning of the exchange-correlation energy is a difficult one, partly because the DFT definitions of exchange and correlation are not exactly the same as those used in wave-function methods. As mentioned in the previous section, electron correlation arises from the correlated behavior between electrons that is not accounted for in the mean-field Hartree-Fock approach. The exchange energy is the total electron-electron repulsion minus the Coulomb repulsion, and is basically a consequence of the Pauli principle, which states that no two electrons can have the same quantum numbers, i. e. two electrons in the same orbital must have opposite spin. 

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