Schrodinger equation molecular wavefunction

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

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. 

In the MO approach molecular orbitals are expressed as a linear combination of atomic orbitals (LCAO) atomic orbitals (AO), in return, are determined from the approximate numerical solution of the electronic Schrodinger equation for each of the parent atoms in the molecule. This is the reason why hydrogen-atom-like wavefunctions continue to be so important in quantum mechanics. Mathematically, MO-LCAO means that the wave-functions of the molecule containing N atoms can be expressed as... 

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. ... 

Spin Orbital. The form of Wavefunction resulting from application of the Hartree-Fock Approximation to the Electronic Schrodinger Equation. Comprises a space part (Molecular Orbital) and one of... 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrodinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrodinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

The basis of ab initio modeling of materials is the time-independent Schrodinger equation in which the state of a molecular system is described with a wavefunction ... 

For energies below the dissociation threshold we can use various coordinate systems to solve the nuclear Schrodinger equation (2.32). If the displacement from equilibrium is small, normal coordinates are most appropriate (Wilson, Decius, and, Cross 1955 ch.2 Weissbluth 1978 ch.27 Daudel et al. 1983 ch.7 Atkins 1983 ch.ll). However, if the vibrational amplitudes increase so-called local coordinates become more advantageous (Child and Halonen 1984 Child 1985 Halonen 1989). Eventually, the molecular vibration becomes unbound and the molecule dissociates. Under such circumstances, Jacobi or so-called scattering coordinates are the most suitable coordinates they facilitate the definition of the boundary conditions of the continuum wavefunctions at infinite distances which we need to determine scattering or dissociation cross sections (Child 1991 ch.l0). Normal coordinates become less and less appropriate if the vibrational amplitudes increase they are completely impractical for the description of unbound motion in the continuum. 

According to Equation (2.29), in the adiabatic representation (index a) one expands the total molecular wavefunction F(R, r, q) in terms of the Born-Oppenheimer states Ej (q R, r) which solve the electronic Schrodinger equation (2.30) for fixed nuclear configuration (R,r). In this representation, the electronic Hamiltonian is diagonal,... 

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. 

Within the Born-Oppenheimer approximation, the Schrodinger equation for a whole molecular system can be divided into two equations. The electronic Schrodinger equation needs to be solved separately for each different (fixed) set of positions for the nuclei making up the system and gives the electronic wavefunction and the electronic... 

Semiempirical methods - based on approximate solutions of the Schrodinger equation with appeal to fitting to experiment (i.e. using parameterization) Density functional theory (DFT) methods - based on approximate solutions of the Schrodinger equation, bypassing the wavefunction that is a central feature of ab initio and semiempirical methods Molecular dynamics methods study molecules in motion. 

The eigenvalue formulation of the Schrodinger equation is the starting point for our derivation of the Hiickel method. We will apply Eq. 4.36 to molecules, so in this context H and i// are the molecular Hamiltonian and wavefunction, respectively. 

Label these statements true or false (1) For each molecular wavefunction there is an electron density function. (2) Since the electron density function has only x, y, z as its variables, DFT necessarily ignores spin. (3) DFT is good for transition metal compounds because it has been specifically parameterized to handle them. (4) In the limit of a sufficiently big basis set, a DFT calculation represents an exact solution of the Schrodinger equation. (5) The use of very big basis sets is essential with DFT. (6) A major problem in density functional theory is the prescription for going from the molecular electron density function to the energy. 

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