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From the partitioning technique, a formulation has been developed that suggests certain linear combinations of functions in an n-particle space as being important contributors to the energy. Using these combinations allows a drastic reduction in the number of functions that are necessary to give the energy to a desired accuracy; a fact which has implications for full CI calculations. These features are illustrated by some results obtained from full CI studies on H2 and HeH+. In the examples studied, it is found that the first-order combination, which corresponds to a “geometric” approximation, gives about two-thirds of the energy that is obtainable from a given basis set. The geometric sumrule is shown to also have other useful properties.

Irreversibility is studied from the point of view of a time-dependent partitioning technique, based on retarded and advanced evolution operators. Application to the Stark effect in the hydrogen atom showed that correct numerical results for the real part of the resonance, that is, the complex pole of the Green's function, could be obtained. (GHT)

It is demonstrated, from general physical principles, that a new kind of quantum correlations, associated with the appropriate reduced density matrix, Γ, exist in amorphous condensed matter. The treatment starts from general representation properties of the second-order reduced density matrix. The derivations, using general statistical notions in a quantum mechanical framework, lead to a decomposition of the two-matrix into the large-, the small-, and the unpaired components. Since this result can be identified with the extreme case of Coleman, the formulation is automatically ensemble-representable. Introducing the complex scaling method (CSM) into the Bloch–Liouville equation for Γ it is shown how the temperature T in the aforementioned ensemble can be specified. This defines, in the present development, thermalization in a system far from equilibrium. It is further shown that this thermalization leads to the formation of certain irreducible (Jordan) blocks containing some (incomplete) phase information. The connection between these irreducible blocks and the concept of a coherent-dissipative structure is made. The treatment reveals general features of the coherent-dissipative structure and a certain universality of the coherence-breaking mechanism. The main equations relate the relaxation time Trcl and the temperature T with the minimum degrees of freedom Smin and the associated spatial domain dmin of the structure. Relevant examples are, e.g., proton transfer and H+ conductivity in water, conductivity of molten alkali chlorides, quantum correlation effects in high-Tc superconductors, and spin dynamics of Gd far above the Curie point.

Simon's exterior-scaling procedure is applied to model systems within the framework of the finite-basis-set approximation. We show that if the basis-set functions are scaled when rg${r}_{0}$, a variational solution is obtained by adding the term 1/2\ensuremath{\delta}(r-${r}_{0}$)[d/dr-\ensuremath{\lambda}] [1-exp(-2i\ensuremath{\Theta})] to the Hamiltonian, where \ensuremath{\lambda} is the logarithmic derivative of the resonance wave function at ${r}_{0}$. Two computational methods are proposed. One is an ab initio linear variational iterative procedure, whereas in the second one \ensuremath{\lambda} is semiclassically estimated during the iteration procedure and all the variational calculations are carried out for r\ensuremath{\le}${r}_{0}$ and therefore are exactly \ensuremath{\Theta} independent. An illustrative numerical application is presented.

The extended virial theorem, obtained from a Fourier–Laplace transformation of equations of motion, is applied to the phenomenon of predissociation. The underlying resonance energy model rigorously defined by means of the theory of dilation analytic operators prescribes in detail the balance between the various energy contributions in a way analogous to the situation prevailing in a stationary system. The extended virial theorem is applied to predissociation, particularly in connection with the interpretation of the nearly degenerate G1Π ∼ I1Π energy spectrum in SiO. It was previously found for the rovibronic spectra assigned to G1Π – X1Σ+ and I1Π – X1 Σ+, that every G – X band is accompanied by a I – X band. Whereas the I1Π state does not seem to undergo predissociation, the G state shows increasing predissociation with increasing vibrational quantum number. We show that the virial theorem generalized to include continuum phenomena offers an interpretation of the nearly degenerate spectra of SiO as well as of the concomitant isotope shift of SiO16 and SiO18 with respect to both resonance positions and widths.

In connection with the theory of resonance phenomena in many-body scattering systems, some aspects of current spectral theories for general operators are reviewed, including theories leading to complex eigenvalues of nonself-adjoint operators well defined over a perturbed originally self-adjoint domain. Consequently, a generalization of the virial theory is obtained. Various applications are considered, particularly, the problem of a system perturbed by an electric field. The associated Stark Hamiltonian is an unbounded operator, but still numerical results in conjunction with a simple reciprocal relation prompt the fulfillment of a virial sum rule. It is said that the extended virial theorem, based on complex scaling, seems to provide a systematic way of arriving at the resonances. 81 references. (JFP)

The possibility of level crossings is discussed from a general multidimensional partitioning viewpoint. By extending the traditional motion of a self-adjoint Hamiltonian to a self-adjoint analytic family of operators, it is found that level intersections that appear fall into two mutually exclusive categories: the conventional diagonal one corresponding to Jordan blocks of order m = 1, and the nondiagonal one with m greater than or equal to 2. Consequences with respect to some recent examples, such as Longuet-Higgins ''sign-reversing loop'' construction and the /sup 1/PI near degeneracy in SiO, are discussed and examined. 21 references.

An optimum continuous representation technique derived in analogy with the self-consistent harmonic approximation in quantum lattice dynamics is discussed and applied to Lipkin's model. The method is based on the variational principle and corresponds to an infinite summation of an expansion of the appropriate Hamiltonian kernel. The overlap kernel is treated exactly. The actual results indicate the removal of the drawbacks of low order perturbation theory previously employed in this connection. (AIP)

The main point of the present investigation is the direct application of the exterior complex scaling procedure, in combination with numerical integration of the Riccati equation, to a numerically given potential. A brief review of the Titchmarsh-Weyl theory for a singular second-order differential equation, its combination with exterior complex scaling and its connections with scattering theory is presented. The method is applied to the A1Π state of the CH+ ion, and the energies and widths for a number of shape resonances are calculated. The Titchmarsh-Weyl results are compared with a calculation using a program by LeRoy, and also compared with experimental results. Further investigations are made concerning poles of the S-matrix far out in the complex plane for a specific value of the rotational quantum number.

Weyl's eigenvalue theory for ordinary second-order differential equations is discussed for the case of a continuous spectrum. It is demonstrated that the spectral density function obtained from a suitably averaged Green's function, equal to the Weinstein function, can be directly related to the Weyl–Titchmarsh m-function. The explicit connections with scattering theory are derived and it is found that the Weyl and Jost solutions are proportional; the proportionality factor being the reciprocal value of the latter at the origin. The physical interpretation of the complex poles of the spectral density is discussed in relation to Gamow's exponentially growing functions. The advantage of using a formulation that allows for a “perturbation” of boundary conditions is pointed out.

A perturbation procedure for arbitrary many electron systems, which incorporates Pade approximants and inner projections, has been developed within the configuration interaction framework. The method is illustrated by an application to the hydrogen molecule.

The Z-expansion of two-electron systems is analyzed with the Pade technique with emphasis on establishing analytical properties of the function E(Z) formally associated with the power series expansion. The concept of critical point in this connection is stressed. For this sequence it occurs at Zc = 0.911246 with E(Zc) = −0.415184. The structure of E(Z) for Z < Zc is investigated. The use of Pade approximants to extrapolate values of electron affinities is emphasized.

In his series of publications “studies in perturbation theory”, /1–2/ Lowdin introduced the bracketing function $$f(z) = $$ (1) where H is the Hamiltonian, φ the reference function belonging to the Hilbert space h and T the reduced resolvent $$T(z) = P{({z^{ - PHP}})^{ - 1}}P$$ (2) $$p = 1 - 0;{\kern 1pt} 0 = \left| {\phi > = 1.$$ (3)

Influenced by the ideas of Jaynes and Prigogine from the mid-1950s, we present a unified formulation of dynamics and thermodynamics of irreversible processes. Our approach originates in the quantum theory of resonances described by effective Hamiltonians. The concept of effective Hamiltonian is extended to the concept of effective Liouvillian that deals with macroscopic observables and brings insight into the dissipative nonequilibrium thermodynamics. The time-energy/frequency Fourier-Laplace transformation and the use of projectors focus on the variables of interest. The line profiles and dynamics in quantum mechanics are treated on the same footing. The long macroscopic times in statistical physics are derived from short microscopic times by means of hierarchies of effective Liouvillians and perturbation theory in the complex plane. The theory is illustrated on solvable models of quasi-continua and continua related to fluctuations and dissipation and on a model of kinetics of a chemical reaction implying a short-lived (resonance) transition state.

http://www.sciencedirect.com/science/article/pii/S0065327610590063; Organisms have evolved a wide variety of mechanisms to utilize and respond to light. In many cases, the biological response is mediated by structural changes that follow photon absorption. These reactions typically occur at femto- to picosecond timescales. As the relevant time and spatial resolutions are notoriously hard to access experimentally, Molecular dynamics simulations are the method of choice to study such ultrafast processes. In this chapter we review the theoretical concepts of photochemical reactions and present a practical simulation scheme for simulating photochemical reactions in biomolecular systems. In our scheme, a multiconfigurational quantum mechanical description is used to model the electronic rearrangement for those parts of the system that are involved in the absorption. For the remainder, typically consisting of the apoprotein and the solvent, a simple forcefield model suffices. The interactions in the systems are thus computed within a hybrid quantum/classical framework. Forces are calculated on-the-fly, and a diabatic surface-hopping procedure is used to model the excited state decay. The validity of this approach is demonstrated by recent MD simulations on photobiological systems that we also review in this chapter. In addition to providing quantities that are experimentally accessible, such as structural intermediates, fluorescence lifetimes, quantum yields, and spectra, the simulations have also provided information that is much more difficult to measure experimentally, such as reaction mechanisms and the influence of individual amino acid residues