Primitive Wave Function Research Articles

An analysis of the accuracy of an initial value representation surface hopping wave function in the interaction and asymptotic regions

The behavior of an initial value representation surface hopping wave function is examined. Since this method is an initial value representation for the semiclassical solution of the time independent Schrodinger equation for nonadiabatic problems, it has computational advantages over the primitive surface hopping wave function. The primitive wave function has been shown to provide transition probabilities that accurately compare with quantum results for model problems. The analysis presented in this work shows that the multistate initial value representation surface hopping wave function should approach the primitive result in asymptotic regions and provide transition probabilities with the same level of accuracy for scattering problems as the primitive method.

Does the polarization approximation converge for large R to a primitive or a symmetry-adapted wavefunction?

Asymptotic expressions for the branch points that most likely determine the radii of convergence for the polarization approximation of the ground states of H + 2 and H 2 are derived. In both cases the radius of convergence is greater than 1 for all R. The radius of convergence for H + 2 differs from unity by a term asymptotically proportional to R 3e −2 R , for H 2 by a term of the order R 6e −4 R . In view of these results, for large R the convergence of the polarization approximation to the exact ground states is so slow that this expansion behaves as if it were converging to a primitive wavefunction (and to the average of the energies of the g and u states). The Hirschfelder—Silbey expansion is free from this defect.

Interaction energies of diatomic molecules using partial antisymmetry and Hartree-Fock atomic wave functions

We provide a numerical test of a new approach to calculating approximate interatomic interaction energies based on partial antisymmetry (PA) [Adams, Chem. Phys. Lett. 68, 511 (1979)]. We use spin-coupled products of single determinant atomic wave functions to approximate a particular primitive wave function of the diatomic. Three methods are used: (1) the conventional full antisymmetry (FA) approach, in which the primitive wave function is antisymmetrized and the difference of expectation values of the total energies of the diatomic and atomic systems is calculated, (2) the PA approach, in which only some of the terms of the antisymmetrizer are applied to the primitive wave function, and the energy expressions are simplified based on approximations to the PA theorem [Adams, op. cit.], and (3) a hybrid approach based on a combination of assumptions from the first two approaches, which is comparable to the work of Dacre and McWeeny [Proc. Roy. Soc. London A317, 435 (1970)]. Results are compared with accurate potential curves from the literature. Interaction energies were calculated for the X 1Σ+g states of Ne2, Li2, and Na2, the X 1Σ+ states of LiNa and LiF, the x 3Σ+u states of Li2 and Na2, x 7Σ+u, and x 3Σ+ LiNa at several internuclear distances. In all cases but Ne2, the PA and FA interaction energies are much closer to each other than is either to the accurate reference values. In these cases there is thus no significant penalty exacted for the use of PA over FA, even though it is easier to use. By conventional reasoning, neither the PA nor the Dacre–McWeeny approaches should work at short range or for binding molecules, because they truncate expansions of the antisymmetrizer in the energy expression. Our results provide counter examples to change that expectation, thus providing additional evidence in support of an approximate approach based on the PA theorem.

Semiclassical calculation of quantum-mechanical wave functions for a two-dimensional scattering system

The semiclassical theory developed by Maslov and Fedoriuk is used to calculate the wave function for two-dimensional scattering from a Morse potential. The characteristic function S and the density Jacobian J are computed in order to obtain the primitive wave function. The incident part shows distorted plane-wave behavior and the scattered part shows radially outgoing behavior. A uniform approximation gives a wave function that is well-behaved near the caustic.

Reply to comment regarding the paper entitled “is antisymmetrization necessary?”

The major point made is that the use of partial antisymmetrization is not an approximation but an alternative to the use of full antisymmetrization in theories of electronic properties. Another point is that the unsymmetrical primitive wave-function may be calculated directly rather than by resolution into symmetry components. Some results for HHe + are presented.

The ‘‘primitive’’ wave function in the theory of intermolecular interactions

The concept of the primitive wave function for a supermolecule consisting of interacting subsystems is critically analyzed. The distinction between formal and genuine primitive functions is stressed. The concept of uniformly complete basis sets as contrasted to simply complete basis is introduced. Primitive basis sets are defined and shown not to be uniformly complete for the expansion of the supersystem wave function while ’full supersystem basis sets’ are. The conditions are specified under which a supersystem wave function can be decomposed into its ’primitive components’ corresponding to different partitions of the electrons among the subsystems. These primitive components satisfy the Schrödinger equation asymptotically. The matrix representation of the Hamiltonian (both the full supersystem Hamiltonian H and the zeroth order Hamiltonian Ho) in terms of these partitions is analyzed. It is shown that in the standard application of RS-perturbation theory to intermolecular forces (the polarization approximation) the limiting processes λ→1 and R→∞ do not commute, that the λ-series is not uniformly convergent with respect to R and that the wave function to any finite order in λ is genuinely primitive. The symmetrized polarization approximation is justified for the ’coasymptotic ground state’ in certain cases and a ’symmetrized polarization approximation with shifted eigenvalues’ is proposed that connects the lowest eigenvalue of Ho with the physical ground state. A justification of simplified schemes in the region of ’small exchange’ is given and alternative perturbation schemes are discussed. Finally the use of the primitive function in variational treatments is outlined. One advantage is that a genuinely (not a formally) primitive function is uniformly expandable in a primitive basis set.

Exchange perturbation theory for electron scattering. Elastic scattering from hydrogen atoms

We present a new perturbation treatment of electron scattering. It is based on determining a primitive wave function which is nonsymmetric with respect to electron exchange. The theory is developed specifically for elastic scattering from atomic hydrogen but appears generalizable to many-electron atoms and to inelastic processes as well. From the first-order wave function, which can be made square-integrable, one obtains the phase shifts through third order. Calculations with a crude spherical-well initial approximation are carried out for illustrative purposes.

Exchange perturbation theory. IV. Calculations on H2+

We have applied the two localized-wave-function (LW) exchange- perturbation-theories (EPT) which we have proposed, to the 1sσg and 2pσu states of H2+ with the objective of verifying the insights gained from these EPT’s and of testing their accuracy. The one LW EPT determines a primitive wave function identical to that of the Eisenschitz–London EPT through first order, but which differs from it in all higher orders. The other determines a function identical to that of the Hirschfelder–Silbey EPT through first order and through infinite order, but which differs from it through all intermediate orders. We find that in terms of the perturbation expansion of the interaction energy through third order, our EPT’s are as accurate as the original EPT’s to which they are related. The LW EPT’s have the conceptual asset that their primitive wave functions are least distorted from the zero order wave function in a precisely defined sense. We have also calculated interaction energies using the integrals which define the interaction energies in terms of LW’s, and substituting the LW’s approximated by sums through first, second, and third order. The energies generally increase in accuracy as the LW is summed to higher orders. When each order contribution to the LW is multiplied by a weight which is determined to minimize the interaction energy, and the LW is summed through third order, the interaction energy is in error by 0.07% or less for nuclear separations ranging from 1.0 to 10.0 bohr. An examination of the LW’s shows how the optimization procedure works. Other quantities are calculated which show that the LW EPT’s are systematically refinable methods for the calculation of LW’s as well as for the calculation of interaction energies. This is important because LW’S may be used to calculate distinct ’’physical’’ contributions to interaction energies.

Natural states of interacting systems and their use for the calculation of intermolecular forces. II. Natural states in the asymptotic 1/ R expansion

The asymptotic natural states i.e. the natural states of the subsystems in the various terms of the asymptotic expansion of the primitive wavefunction of the supersystem are defined. Integro-differential equations are derived that allow the direct calculation of these asymptotic natural states and from them, the terms C 6, C 8 etc. in the asymptotic 1/ R expansion of the interaction energy. A pictorial interpretation of these states is given and the rapid convergence of the expansion in natural states is explained. The transferability of certain matrix elements is demonstrated and a virtually exact combination rule for the calculation of the C k coefficients between different atoms from those for like atoms is obtained- The effects of intraatomic electron correlation on the C k as well as the justification of a core-valence separation are discussed.

Localization in exchange perturbation theory

A new formalism is developed for exchange perturbation theory by applying a localization criterion to remove much of the arbitrariness associated with the definition of the primitive wavefunction. In contrast to Adams’ method, which is based on a certain energy-related localization, our prescription is based on a direct spatial localization of the primitive function by maximizing its overlap with a given reference function. If our primitive function is constrained only by requiring that one of its symmetry projections yield an exact eigenfunction, then the wavefunction and energy coincide through first order with that of the Eisenschitz–London (EL), and Hirschfelder–van der Avoird (HAV) methods, and through the second order with the Murrell–Shaw–Musher–Amos (MSMA) method, while differing in the third and higher orders. If, instead, the primitive function is constrained by requiring that all of its symmetry projections yield exact eigenfunctions, then it coincides in first order with that of the Distinguishable Electron Method (DEM) developed by Kirtman and co-workers. Our analysis then provides a new justification of the first order terms in the EL–HAV, MSMA, and the DEM expansions, and of the second order terms in the MSMA expansion, and also provides soundly based extensions to higher orders.