Atomic variation principle

From the preceding discussion, the mode of integration used in the definition of an atomic property is determined by the atomic variation principle and is the same as that used in the definition of the charge density itself. The atomic average of an observable A is given by... 

The statement of the atomic action principle given in eqn (8.145) is a variational principle which enables one to derive the properties of an atom in a molecule—it is an atomic variation principle. We shall use it first to derive the atomic statement of the Ehrenfest force law, the equation of motion of an atom in a molecule. This is accomplished through a variation of fl]... 

Thus the atomic variation principle states that... 

By combining the results given in eqns (8.172) and (8.174) one obtains from the atomic variational principle the integrated atomic force law... 

The general time-dependent virial theorem for an atom in a molecule is derived from the atomic variational principle. We shall find a close connection between the expressions so obtained for the virial and those derived in the previous section for the force. In particular, the differential force law leads directly to a corresponding local expression for the virial theorem. 

The average of the atomic virial V,(12), eqn (8.192), as defined by the atomic variational principle, is the virial of the quantum mechanical force density as defined in the differential force law, averaged over the atomic volume. By taking the virial of F(r, t) in eqn (8.179), one obtains... 

ACTION PRINCIPLE FOR A QUANTUM SUBSYSTEM 8.4 8.4.5 Summary of the atomic variational principle... 

For example, F = eN = slN, where N is the total number of electrons in the system, generates a simple infinitesimal transformation, which leaves the Lagrangian 2] invariant. In addition, since is a constant of the motion for the total system, [i , N] = 0. However, the time rate of change of the average electronic population of an atom, N( l), is not zero in general and the equation of continuity governing the time evolution of Al( 2) is obtained directly from the equivalent statement of the atomic variational principle, eqn (8.149), as... 

The representation of trial fiinctions as linear combinations of fixed basis fiinctions is perhaps the most connnon approach used in variational calculations optimization of the coefficients is often said to be an application of tire linear variational principle. Altliough some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend tliese calculations to larger atoms and molecules quickly runs into fonnidable difficulties (not the least of which is how to choose the fomi of the trial fiinction). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize are all easily carried out by computers. 

In the quantum mechanics of atoms and molecules, both perturbation theory and the variational principle are widely used. For some problems, one of the two classes of approach is clearly best suited to the task, and is thus an established choice. Flowever, in many others, the situation is less clear cut, and calculations can be done with either of the methods or a combination of both. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

For the H2 system, the amount of ionic HL structures determined by the variational principle is 44%, close to the MO-HF value of 50%. The need for including large atnounts of ionic structures in the VB formalism is due to the fact that pure atomic orbitals are used. 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

Among various theories of electronic structure, density functional theory (DFT) [1,2] has been the most successful one. This is because of its richness of concepts and at the same time simplicity of its implementation. The new concept that the theory introduces is that the ground-state density of an electronic system contains all the information about the Hamiltonian and therefore all the properties of the system. Further, the theory introduces a variational principle in terms of the ground-state density that leads to an equation to determine this density. Consider the expectation value (H) of the Hamiltonian (atomic units are used)... 

The concept of dipole hardness permit to explore the relation between polarizability and reactivity from first principles. The physical idea is that an atom is more reactive if it is less stable relative to a perturbation (here the external electric field). The atomic stability is measured by the amount of energy we need to induce a dipole. For very small dipoles, this energy is quadratic (first term in Equation 24.19). There is no linear term in Equation 24.19 because the energy is minimum relative to the dipole in the ground state (variational principle). The curvature hi of E(p) is a first measure of the stability and is equal exactly to the inverse of the polarizability. Within the quadratic approximation of E(p), one deduces that a low polarizable atom is expected to be more stable or less reactive as it does in practice. But if the dipole is larger, it might be useful to consider the next perturbation order ... 

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)]. 

The first kind of simplification exclusively concerns the size of the basis set used in the linear combination of one center orbitals. Variational principle is still fulfilled by this type of "ab initio SCF calculation, but the number of functions applied is not as large as necessary to come close to the H. F. limit of the total energy. Most calculations of medium-sized structures consisting for example of some hydrogens and a few second row atoms, are characterized by this deficiency. Although these calculations belong to the class of "ab initio" investigations of molecular structure, basis set effects were shown to be important 54> and unfortunately the number of artificial results due to a limited basis is not too small. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

Our object of interest is a many electron finite system (such as an atom, molecule, cluster etc.), having, by assumption, a nondegenerate ground state (GS) (this assumption will be removed in Sects. 4.4 and 5). The numter of electrons N and the electron-nuclei potential energy v(r) = Ve (r) (the so-called external potential) are given and common for all schemes to be discussed. The GS energy qs aod the GS wave function Vqs of the system can be found from a variational principle as... 

To get some idea of the use of trial wave functions and the variation principle, evaluate the expectation value of the energy using the Hydrogen atom Hamiltonian, and normalized Is orbitals with variable Z. That is, evaluate ... 

The existence of many ionic structures in MCVB wave functions has often been criticized by some workers as being unphysical. This has been the case particularly when a covalent bond between like atoms is being represented. Nevertheless, we have seen in Chapter 2 that ionic structures contribute to electron delocalization in the H2 molecule and would be expected to do likewise in all cases. Later in this chapter we will see that they can also be interpreted as contributions from ionic states of the constituent atoms. When the bond is between unlike atoms, it is to be expected that ionic stmctures in the wave function will also contribute to various electric moments, the dipole moment being the simplest. The amounts of these ionic structures in the wave functions will be determined by a sort of balancing act in the variation principle between the diagonal effects of the ionic state energies and the off-diagonal effect of the delocalization. 

As an example we apply the variational principle to the evaluation of the ground state energy of a hydrogen-like atom using a minimum basis set of two-component radial functions ... 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

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