Schrodinger equation Slater determinants

Without these terms the Schrodinger equation can be solved exactly, with the solution being a Slater determinant composed of orbitals. 

In addition to the Schrodinger equation we have the antisymmetry requirement (Eq. II.2) connected with the Pauli principle and, by means of the antisymmetrization operator (Eq. 11.16), the Hartree product (Eq. 11.37) is then transformed into a Slater determinant ... 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

One of the more radical approximations introduced in the deduction of the Hartree-Fock equations 2 from the Schrodinger equation 3 is the assumption that the wavefunction can be expressed as a single Slater determinant, an antisymmetrized product of molecular orbitals. This is not exact, because the correct wavefunction is in fact a linear combination of Slater determinants, as shown in equation 5, where Di are Slater determinants and c are the coefficients indicating their relative weight in the wavefunction. 

Wia is the Slater determinant in which the i occupied spinorbital is replaced by the a unoccupied one. Setting this ansatz in the Schrodinger equation, one obtains a set of linear equations... 

The electronic Schrodinger equation is still intractable and further approximations are required. The most obvious is to insist that electrons move independently of each other. In practice, individual electrons are confined to functions termed molecular orbitals, each of which is determined by assuming that the electron is moving within an average field of all the other electrons. The total wavefunction is written in the form of a single determinant (a so-called Slater determinant). This means that it is antisymmetric upon interchange of electron coordinates. ... 

For the very small systems in Table 7.1, it is possible to approach the exact solution of the Schrodinger equation, but, as a rule, convergence of the correlation energy is depressingly slow. Mathematically, this derives from the poor ability of products of one-electron basis functions, which is what Slater determinants are, to describe the cusps in two-electron densities that characterize electronic structure. For the MP2 level of theory, Schwartz (1962)... 

Equation (11.14) is an example of a non-linear Schrodinger equation. It can be solved in the usual HF fashion by constmction of a Slater determinant formed from MOs i/f that are optimized using a modified Fock operator according to... 

Based on first principles. Used for rigorous quantum chemistry, i. e., for MO calculations based on Slater determinants. Generally, the Schrodinger equation (Hy/ = Ey/) is solved in the BO approximation (see Born-Oppenheimer approximation) with a large but finite basis set of atomic orbitals (for example, STO-3G, Hartree-Fock with configuration interaction). 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

HF is the simplest of the ab initio methods, named after the fact that they provide approximate solutions to the electronic Schrodinger equation without the use of empirical parameters. More accurate, correlated, ab initio methods use an approximate form for the wavefunction that goes beyond the single Slater determinant used in HF theory, in that the wavefunction is approximated instead as a combination or mixture of several Slater determinants corresponding to different occupation patterns (or configurations) of the electrons in the molecular orbitals. When an optimum mixture of all possible configurations of the electrons is used, one obtains an exact solution to the electronic Schrodinger equation. This is, however, not computationally tractable. 

QM grew out of studies of blackbody radiation and of the photoelectric effect. Besides QM, radioactivity and relativity contributed to the transition from classical to modem physics. The classical Rutherford nuclear atom, the Bohr atom, and the Schrodinger wave-mechanical atom are discussed. Hybridization, wavefunctions, Slater determinants and other basic concepts are explained. For obtaining eigenvectors and eigenvalues from the secular equations the elegant and simple matrix diagonalization method is explained and used. All the necessary mathematics is explained. 

The derivation of the Roothaan-Hall equations involves some key concepts Slater determinant, Schrodinger equation, explicit Hamiltonian operator,... 

Beyond the Molecular Orbital Approach Introduction.—In principle an exact solution to the non-relativistic Schrodinger equations for a molecule can be achieved by the configuration interaction technique. A complete set of one-electron spin orbitals i is used to form a complete set of Slater determinants by choosing all possible ordered sets of n elements of the set of 4u s. A linear combination of these determinants is then used ... 

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