First principles semiclassical calculations of vibrational eigenfunctions

Vibrational eigenfunctions are calculated on-the-fly using semiclassical methods in conjunction with ab initio density functional theory classical trajectories. Various semiclassical approximations based on the time-dependent representation of the eigenfunctions are tested on an analytical potential describing the chemisorption of CO on Cu(100). Then, first principles semiclassical vibrational eigenfunctions are calculated for the CO(2) molecule and its accuracy evaluated. The multiple coherent states initial value representations semiclassical method recently developed by us has shown with only six ab initio trajectories to evaluate eigenvalues and eigenfunctions at the accuracy level of thousands trajectory semiclassical initial value representation simulations.     

Contracted basis Lanczos methods for computing numerically exact rovibrational levels of methane

We present a numerically exact calculation of rovibrational levels of a five-atom molecule. Two contracted basis Lanczos strategies are proposed. The first and preferred strategy is a two-stage contraction. Products of eigenfunctions of a four-dimensional (4D) stretch problem and eigenfunctions of 5D bend-rotation problems, one for each K, are used as basis functions for computing eigenfunctions and eigenvalues (for each K) of the Hamiltonian without the Coriolis coupling term, denoted H0. Finally, energy levels of the full Hamiltonian are calculated in a basis of the eigenfunctions of H0. The second strategy is a one-stage contraction in which energy levels of the full Hamiltonian are computed in the product contracted basis (without first computing eigenfunctions of H0). The two-stage contraction strategy, albeit more complicated, has the crucial advantage that it is trivial to parallelize the calculation so that the CPU and memory costs are independent of J. For the one-stage contraction strategy the CPU and memory costs of the difficult part of the calculation scale linearly with J. We use the polar coordinates associated with orthogonal Radau vectors and spherical harmonic type rovibrational basis functions. A parity-adapted rovibrational basis suitable for a five-atom molecule is proposed and employed to obtain bend-rotation eigenfunctions in the first step of both contraction methods. The effectiveness of the two methods is demonstrated by calculating a large number of converged J = 1 rovibrational levels of methane using a global potential energy surface.     

Normalization of projected spin eigenfunctions

We find the normalization integral for projected spin eigenfunctions, defined by means of character projection operators of the symmetric group. We also obtain a reduced expression for these spin eigenfunctions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Laplace transform wave functions

The wave function defining a quantum-mechanical system is considered as the Laplace transform of some distribution and the consequent form of the Variational Principle derived; an integral equation defines the eigenfunctions of a certain subclass. The model of the hydrogen-like atom is used to test the theory; the eigenfunctions and associated energy levels of the ground and excited states are obtained for arbitrary values of the orbital quantum number.     

Variational method for calculating the anharmonic vibrations of polyatomic molecules using a combined morse-anharmonic basis taking into account the fragmentary structure of molecules

A method of variational solution of anharmonic vibration problems using a mixed Morse—anharmonic basis is proposed. The basis functions are the products of the Morse oscillator eigenfunctions for vibrations of peripheral bonds, the harmonic oscillator eigenfunctions for almost harmonic skeletal and deformation vibrations, and the anharmonic basis functions for essentially anharmonic skeletal and deformation vibrations. The anharmonic basis wave functions are taken as a linear combination of the Morse and harmonic oscillator eigenfunctions. The introduction of the combined Morse—anharmonic functions allows one to factorize the solution of a problem into a series of individual blocks according to the fragmentary structure of molecules. Volgograd Pedagogical University. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 231–238, March–April, 1995. Translated by I. Izvekova     

Eigenvalues of the Boltzmann collision operator for binary gases: Mass dependence

Eigenvalues of the Boltzmann collision operator are calculated versus mass ratio with two different methods. One method involves the expansion of the eigenfunctions in speed polynomials, whereas with the second method the eigenfunctions are evaluated at discrete points based on a particular gaussian quadrature rule. The discrete ordinate method proved to be superior provided the mass ratio was neither too large nor too small. The approach of the eigenvalues to the continuum boundary was also studied for several mass ratios.     

A simple algorithm for the construction of genealogical spin functions

A simple algorithm is given for the construction of spin eigenfunctions according to the genealogical scheme. The method can deal directly with the N-electron problem without any knowledge of the (N-1)-electron spin eigenfunctions. It uses the representation matrices corresponding to the transpositions (k,k + 1), the latter can be written down from the knowledge of the Young tableaux.     

SPINS: A collection of algorithms for symbolic generation and transformation of many-electron spin eigenfunctions

Summary SPINS represents a collection of algorithms intended to provide an efficient, robust and easy-to-use quantum-chemical toolbox capable of performing a wide range of operations on spin eigenfunctions in the Rumer, Kotani and Serber spin bases. It includes routines for symbolic generation of the Rumer spin eigenfunctions as linear combinations of elementary spin products, for computing all transformation matrices relating the Rumer, Kotani and Serber spin bases and for calculation of the matrices of the irreducible representations of the symmetric group carried by the Rumer, Kotani and Serber spin eigenfunctions, as well as facilities for interpreting general spin-coupling patterns such as those used in spin-coupled theory. The resulting codes, written in Fortran-77 and available on the Internet (from P.B.Karadakov@Bristol.AC.UK or DLC@Liverpool.AC.UK) are so compact and efficient that they even run on IBM PC-compatible personal computers.     

Incompletely symmetric non-bloch electron states in perfect cubic crystals. III. Density of dynamical observables in the FCC lattice

The operators of dynamical observables of the crystal electron (velocity vector components. reciprocal mass tensor components and their functions) commute with the energy operator; hence, the averages of these observables can be adequately approximated by the eigenfunctions for the energy operator. Calculations of the averages were based on the LCAO eigenfunctions classified according to incompletely symmetric irreducible representations of the point group of the cubic crystal, and a similar classification was made for the averages.     

Collisional energy transfer with statistical energy exchange: an analytical solution in the statistical limit

Collisional energy transfer between highly vibrationally excited molecules and bath gas is considered with a statistical kernel, describing energy exchange in complex-forming collisions. Knowledge of the bilinear formula for the Laguerre polynomials offers a means for determining eigenvalues and eigenfunctions of the kernel. An exact solution to master equation for the conditional probability is given as an expansion in terms of these eigenfunctions. The bulk averages of internal energy moments and energy transfer moments are calculated analytically.     

Filter Diagonalization with Finite Fourier Transform Eigenfunctions

Below, we briefly report on the progress in the development of the Filter Diagonalization technique when filtering is carried out with the aid of Finite Fourier Transform (FFT) eigenfunctions. During recent years interest in these functions, also known as ‘prolates’, or ‘slepians’, has increased among scientists doing research in the field of signal processing. The main explanation to this follows from the set of very special extremal and orthogonality properties exibited by the FFT eigenfunctions. Recent results of Walter and Shen on sampling with prolate spheroidal functions will necessary produce a new wave of interest. In the presented, Filter diagonalization machinery, we show that the sampling formula of Walter and Shen simplifies essentially the computation of matrix elements as certain 2D–integrals involving FFT eigenfunctions.     

Generation of genealogical spin eigenfunctions

A method is given for generating the Yamanouchi-Kotani genealogical spin eigenfunctions which requires neither storage of eigenfunctions for smaller numbers of electrons, nor summations of large order, nor explicit use of results from the theory of representations of the symmetric group. An explicit formula is given for the coefficients of expansion in terms of spin products.     

Incompleteness of atomic and molecular eigenfunctions

It is well known that the set of bound state hydrogenic eigenfunctions is not complete. Here we show that the set of bound state eigenfunctions for all atomic and molecular systems is incomplete in the Born–Oppenheimer approximation. A trivial argument that there are no positive eigenvalues is also presented which results from the virial theorem. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the random walk of energy to traps in disordered systems

Adams recently derived a theorem concerning solutions of the Schrödinger equation for an N-electron system. From this theorem he concludes that exact eigenvalues and (antisymmetric) eigenfunctions can be obtained by solving a different type of equation whose eigenfunctions are not antisymmetric. We critically discuss aspects of Adams' formalism.     

Quasi-algebraic treatment of bound states of simple one-electron systems

We present a straightforward, quasi-algebraic treatment of simple one-particle quantum-mechanical systems. The method consists primarily of a canonical transformation that changes the Schrödinger equation into a first-order differential equation, thus allowing an easier derivation of the eigenvalues and eigenfunctions. We express the latter in a way which is not commonly encountered in the standard literature on quantum mechanics and quantum chemistry. The derivation of generating functions for the eigenfunctions offers no difficulty because the method is formulated in the coordinate representation. As illustrative examples, we consider the harmonic oscillator and a particle in a Kratzer potential.     

On the eigenfunctions of the smoluchowski diffusion equation in a double-well potential and their relation to polymer dynamics

The eigenspectrum, matrix elements and sum-rule contributions for a Smoluchowski equation in a potential U = U₀(x²?1)² are obtained. The importance of the higher eigenfunctions and the sum rule is discussed and it is shown that a reasonably complete solution can be obtained with a comparatively small number of eigenfunctions.     

Extended states of a shallow donor located near a semiconductor–insulator interface

Scattering of a conduction electron by a charged shallow donor located near a semiconductor–insulator interface in the semiconductor or by a charged center embedded in the insulator is considered within the model of a hydrogenlike atom in a semi-infinite space. The interface influence is allowed for by spatial confinement of the electron envelope wave function. The impurity electrostatic image at the interface is taken into account. The problem is separable in prolate spheroidal coordinates and thus is solvable exactly. A rapidly convergent expansion is proposed for the angular eigenfunctions. The radial eigenfunctions are calculated directly by numerical integration of the radial boundary value problem. Expansions of the scattering wave function and the scattering amplitude in terms of the eigenfunctions of the problem are obtained. Using the extended and localized state wave functions, the photoionization cross section of a shallow donor near a semiconductor–insulator interface is calculated. It is presented as a superposition of the oscillator strengths of transitions to the partial extended eigenstates that constitute the scattering wave function. Near the interface, the cross section is enhanced significantly and redistributed over the direction of photoionized electron escape. The photoionization threshold follows the localized state energy varying with the donor–interface distance. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 435–456, 1998     

Comparison of two genres for linear scaling in density functional theory: purification and density matrix minimization methods

Two classes of linear-scaling methods to replace diagonalization of the one-particle Hamiltonian matrix in density functional theory are compared to each other. Purification takes a density matrix with the correct eigenfunctions and corrects the occupation numbers; density matrix minimization takes a density matrix with correct occupation numbers and corrects the eigenfunctions by rotating the orbitals. Computational comparisons are performed through modification of the MondoSCF program on water clusters and the protein endothelin. A purification scheme and a density matrix minimization scheme, based on the 1,2-contracted Schrodinger equation [D. A. Mazziotti, J. Chem. Phys. 115, 8305 (2001)] are implemented in large systems.