Atomic electron wavefunction

Laguerte polynomials are one factor in the radial part of the one-electron atom electronic wavefunctions the other factor is the exponential envelope... 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

I ain going to leave you to prove for yourself that the wavefunction corresponding to this infinite-distance H2 problem is a product of two hydrogen atom wavefunc-tions. Physically, you might have expected this the two atoms are independent so the electronic wavefunctions multiply to give the molecular electronic... 

If I write possible atomic orbitals for hydrogen atom A as Xa possible atomic orbitals for hydrogen atom B as the molecular electronic wavefunction will be... 

Where might these one-electron wavefunctions come from I explained the basic ideas of HF and HF-LCAO theory in Chapter 6 we could find the molecular orbitals as linear combinations of appropriate atomic orbitals by solving the HF eigenvalue problem... 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

Ab initio calculations usually begin with a solution of the Hartree-Fock equations, which assumes the electronic wavefunction can be written as a single determinant of molecular orbitals. The orbitals are described in terms of a basis set of atomic functions and the reliability of the calculation depends on the quality of the basis set being used. Basis sets have been developed over the years to produce reliable results with a minimum of computational cost. For example, double zeta valence basis sets such as 3-21G [15] 4-31G [16] and 6-31G [17] describe each atom in the molecule with a single core Is function and two functions for the valence s and p functions. Such basis sets are commonly used, as there appears to be a cancellation of errors, which fortuitously allows them to predict quite accurate results. 

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 ... 

Applying the permutation operator P12 is therefore equivalent to interchanging rows of the determinant in Eq. (2.15). Having devised a method for constructing many-electron wavefunctions as a product of MOs, the final problem concerns the form of the many-electron Hamiltonian which contains terms describing the interaction of a given electron with (a) the fixed atomic nuclei and (b) the remaining (N— 1) electrons. The first step is therefore to decompose H(l, 2, 3,..., N) into a sum of operators Hj and H2, where ... 

P has been computed using Hartree-Fock atomic orbital wavefunctions and can be found in several published tabulations14 17 and in Appendix 1. Because of the (r 3) dependence of P, dipolar coupling of a nuclear spin with electron spin density on another atom is usually negligible. 

Figure 3.2 Overlap of atomic ls-wavefunctions of H-atoms to form an H2 molecule with an electron-pair bond.

In the foregoing, U is the interaction potential, M is the reduced mass of the colliding system, ftk and ftk are respectively the momentum of the projectile before and after the collision, ig and in are respectively the wavefunctions of the atom (or molecule) in the ground and nth excited states, and the volume element dt includes the atomic electron and the projectile. Since U for charged-particle impact may be represented by a sum of coulombic terms in most cases, Eq. (4.11) can be written as (Bethe, 1930 Inokuti, 1971)... 

In analogy to using a linear combination of atomic orbitals to form MOs, a variational procedure is used to construct many-electron wavefunctions from a set of N Slater determinants y, i.e. one sets up a N x. N matrix of elements flij = (d>, H d>y) which, upon diagonalization, yields state energies and associated vectors of coefficients a used to define (fi as a linear combination of A,s ... 

The essence of Schrodinger s treatment was to replace the classical orbit of Bohr s semi-classical (particle) model of the H-atom by a corresponding wavelike orbital (single-electron wavefunction) L. Instead of specifying the electron s... 

How can one join an electronic structure calculation with a classical MD scheme In principle, this is possible in a straightforward manner - we can optimize the electronic wavefunction for a given initial atomic configuration (at time t=0) and calculate the forces acting on the atoms via the Hellman-Feynman theorem ... 

Warshel and Levitfs study of lysozyme initiated the field of QM/ MM methods [29]. Many groups [10, 30-33] have covered the basic theory of QM/MM methods in detail, so we will outline the theoretical basis only briefly. In accordance with Field et al. [30], the energy of the whole system, E, is written in terms of an effective Hamiltonian, Heff, and the electronic wavefunction of the QM atoms, y/ ... 

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