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  • 1
    Electronic Resource
    Electronic Resource
    The problem of unphysical states in the theory of intermolecular interactions (1992)
    Springer
    Journal of mathematical chemistry 10 (1992), S. 1-23 
    Details
    ISSN: 1572-8897
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Mathematics
    Notes: Abstract It was shown by Claverie that the interactions between atoms and molecules make unphysical electronic solutions of the Schradinger equation accessible in perturbation calculations of intermolecular interactions, accessible in the sense that the perturbation expansion is likely to converge to an unphysical solution if it converges at all. This is a difficult problem because there are generally an infinite number of unphysical states with energies below that of the physical ground state. We have carried out configuration interaction calculations on LiH of both physical and unphysical states. They show that avoided crossings occur between physical and unphysical energy levels as the interaction between the two atoms is turned on, i.e. as the expansion parameter λ is increased from 0 to 1. The avoided crossing for the lowest energy state occurs for λ 〈 0.8, implying that the perturbation expansion will diverge for larger values of λ. The behavior of the energy levels as functions of λ. is shown to be understandable in terms of a two-state model. In the remainder of the paper, we concentrate on designing effective Hamiltonians which have physical solutions identical to those of the Schrödinger equation, but which have no unphysical states of lower energy than the physical ground state. We find that we must incorporate ideas from the Hirschfelder-Silbey perturbation theory, as modified by Polymeropoulos and Adams, to arrive finally at an effective Hamiltonian which promises to have the desired properties, namely, that all unphysical states be higher in energy than the physical bound states, that the first and higher order corrections to the energy vanish in the limitR = ∞. that the leading terms of the asymptotic 1/R expansion of the energy be given correctly in second order, and that the overlap between the zeroth order wave function and the corresponding eigenfunction of the effective Hamiltonian be close to one.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01169168
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  • 2
    Electronic Resource
    Electronic Resource
    Perturbation theory of intermolecular interactions: What is the problem, are there solutions? (1990)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 38 (1990), S. 531-547 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: We review the nature of the problem in the framework of Rayleigh-Schrödinger perturbation theory (the polarization approximation) considering explicitly two examples: the interaction of two hydrogen atoms and the interaction of Li with H. We show, in agreement with the work of Claverie and of Morgan and Simon, that the LiH problem is dramatically different from the H2 problem. In particular, the physical states of LiH are higher in energy than an infinite number of discrete, unphysical states and they are buried in a continuum of unbound, unphysical states, which starts well below the lowest physical state. Claverie has shown that the perturbation expansion, under these circumstances, is likely to converge to an unphysical state of lower energy than the physical ground state, if it converges at all. We review, also, the application of two classes of exchange perturbation theory to LiH and larger systems. We show that the spectra of three Eisenschitz-London (EL) class, exchange perturbation theories have no continuum of unphysical states overlaying the physical states and no discrete, unphysical states below the lowest physical state. In contrast, the spectra of two Hirschfelder-Silbey class theories differ hardly at all from that found with the polarization approximation. Not one of the EL class of perturbation theories, however, eliminates all of the discrete unphysical states. The best one establishes a one-to-one correspondence between the lowest energy states of the unperturbed and perturbed Hamiltonians, and a one-to-two correspondence for the higher states. We suggest that the EL class perturbation theories would be good starting points for the development of more effective perturbation theories for intermolecular interactions.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560382452
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  • 3
    Electronic Resource
    Electronic Resource
    Interaction energies of diatomic molecules using partial antisymmetry and Hartree-Fock atomic wave functions (1990)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 38 (1990), S. 753-765 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: 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.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560382474
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  • 4
    Electronic Resource
    Electronic Resource
    Perturbation theory of intermolecular interactions: Are second-order Rayleigh-Schrödinger energies meaningful? (1991)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 40 (1991), S. 165-181 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: The interactions between atoms are so strong that they create an infinite number of unphysical states of lower energy than the physical ground state, thus raising the question of the significance of second and higher order perturbation energies. We have carried out a series of configuration interaction calculations which shed light on this question and other questions regarding the perturbation theory of intermolecular interactions. The calculations used an unconventional set of configuration functions, each being the product of two Slater determinants, one for each atom. This makes it possible to calculate approximations to both unphysical and physical states, and to follow the changes in eigenstate energies as the interaction is turned on between atoms. We report results for LiH which show that eigenvalues of Ĥ0 + λV̂, where Ĥ0 is the sum of atomic Hamiltonians and V̂ the interaction potential, undergo avoided crossings between λ = 0 and 1. The perturbation expansion of the ground state energy must become divergent at the first avoided crossing, λ = 0.76. In addition, we report results of calculations with three effective Hamiltonians which have all unphysical state energies shifted into the physical continuum. They show no avoided crossings between the ground state and other states until λ is almost equal to 1. The perturbation expansion derived from one of these effective Hamiltonians is shown to have several desirable properties which the expansions derived from the other two lack. Its one fault is that it must diverge at λ = 1, where an avoided crossing occurs. This avoided crossing is related to the exchange symmetry of electrons. The implications of our results for the meaning of second-order perturbation energies and for the development of convergent exchange perturbation methods is discussed.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560400819
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