Ground-state electronic wave function

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

In Table II we also compare our total variational energies with the energies obtained by Wolniewicz. In his calculations Wolniewicz employed an approach wherein the zeroth order the adiabatic approximation for the wave function was used (i.e., the wave function is a product of the ground-state electronic wave function and a vibrational wave function) and he calculated the nonadiabatic effects as corrections [107, 108]. In general the agreement between our results... 

The separability used here leads to a clear relationship between chemical species and ground state electronic wave functions. Each isomeric species is determined by its own stationary ground state electronic wave function. The latter determines a stationary arrangement of Coulomb sources which is different for the different isomers. The nuclei are then hold around a stationary configuration if eq.(10) has bound solutions. An interconversion between them would require a Franck-Condon process, as it is discussed in Section 4. 

Since the exact ground-state electronic wave function and density can only be approximated for most A-electron systems, a variational theory is needed for the practical case exemplified by an orbital functional theory. As shown in Section 5.1, any rule T 4> defines an orbital functional theory that in principle is exact for ground states. The reference state for any A-electron wave function T determines an orbital energy functional E = Eq + Ec,in which E0 = T + Eh + Ex + V is a sum of explicit orbital functionals, and If is aresidual correlation energy functional. In practice, the combination of exchange and correlation energy is approximated by an orbital functional Exc. 

High level ab initio calculations have been described for CuO [63,64], The ground state electronic wave function is fairly complicated, but the contributing configuration which gives rise to the 63 Cu hyperfine interaction may be written. 

In the expressions above t/)c(Ro H/j) is the ground state electronic wave function at the equilibrium nuclear position Rq, X are the Cartesian displacement coordinates of nucleus A(q = x,y,z), related to the normal coordinates by S ai. 

R and r denote nuclear and electronic coordinates, respectively, and Zn is the charge on nucleus N. JCei is the usual electronic Hamiltonian for fixed positions of the nuclei, ei (r R) is the many-electron ground state electronic wave function that depends parametrically on the nuclear positions, R, and and represent sums over electrons and / ... 

The reader familiar with the historical development of quantum mechanics can be forgiven if they greet with skepticism the notion that one can extract all the information contained in a ground-state electronic wave function from the probability distribution function for observing an electron at the point r (that is, the ground-state electron density). The first hint that such a construction might be possible follows directly from the form of the molecular Hamiltonian, Eq. (2). Consider that the form of the kinetic energy operator... 

Consider what the H2 wave function would look like for large values of the inter-nuclear separation R. When the electron is neeu nucleus a, nucleus b is so far away that we essentially have a hydrogen atom with origin at a. Thus, when r is small, the ground-state electronic wave function should resemble the ground-state hydrogen-atom wave function of Eq. (6.104). We have Z = 1, and the Bohr radius Oq has the numerical value 1 in atomic units hence (6.104) becomes... 

Before solving (8.57), let us improve the trial function (13.41). Consider the limiting behavior of the H2 ground-state electronic wave function as R goes to zero. In this limit we get the He" ion, which has the ground-state wave function [put Z = 2 in (6.104)]... 

The proof of the Hohenberg-Kohn theorem is as follows. The ground-state electronic wave function ipQ of an n-electron molecule is an eigenfunction of the purely electronic Hamiltonian of Eq. (13.5), which, in atomic units, is... 

The ground-state electronic wave function of QHg belongs to the totally symmetric symmetry species and is an eigenfunction of with eigenvalue -i-l.The functions I and II combine with equal coefficients to form a single symmetry function, and the functions III, IV, and V combine to form a second symmetry function. 

In the perturbational approach (cf. 232) to the electron correlation, the Hartree-Fock function, >0, is treated as the zero-order approximation to the true ground-state wave function i.e., I>o = Thus, the Hartree-Fock wave function stands at the starting point, while the goal is the exact ground-state electronic wave function. 

To study chemical dynamics, it is often sufficient to restrict the electronic basis to the ground state electronic wave function, ( o(r> R), and this choice will be assumed throughout this work. The associated electronic energy, o(R)> is seen to depend on the particular structure of the molecule, R. This dependence defines the topology of the potential energy landscape on which the nuclear dynamics happens. 

Approaches for tackling the many-body problem based on the density have been around since the 1920s [10, 11]. The birth of modern DFT for the electronic structure problem, however, came with the realization and associated proof by Hohenberg and Kohn in 1964 [12] that the ground-state electronic wave function, vko, is a unique functional of the ground-state electron density, no, i.e.. 

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