The quasiparticle concept in vibrational–electronic problems in molecules. I. Partitioning of the vibrational–electronic Hamiltonian

The partitioning of the vibrational–electronic Hamiltonian is presented. This partitioning is based on a new quasiparticle transformation that is constructed in such a way that the adiabatic approximation is included into the unperturbed Hamiltonian; nonadiabacity, anharmonicity, and electron correlation are treated as perturbations. We also present the second quantization treatment for bosons. The many body perturbation theory expansion for the vibrational–electronic Hamiltonian is suggested. A comparison of this approach is made with gradient techniques.     

Geminal approach to vibronic models of superconductivity in crystals. I. The Jahn–Teller effect

Electronic geminals constructed as linear combinations of binary products of site functions are used to formulate a vibronic model of superconductivity in crystals that is based upon the approximation of independent correlated electron pairs obtained variationally from an electron‐pair Hamiltonian and the Jahn–Teller effect. The cyclic symmetry of the system is taken into account and the geminals are sorted into doubly degenerate pairs. The Herzberg–Teller expansion of the pair Hamiltonian in terms of vibrational modes leads directly to the Jahn–Teller effect. A contact transformation of the vibronic Hamiltonian containing only linear terms lowers the energy of the system by a second‐order term associated with the Jahn–Teller stabilization energy. A possible model for superconductivity in solids is proposed on the basis of the Jahn–Teller effect. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Multi‐state multi‐reference Møller–Plesset second‐order perturbation theory for molecular calculations

This work presents multi‐state multi‐reference Møller–Plesset second‐order perturbation theory as a variant of multi‐reference perturbation theory to treat electron correlation in molecules. An effective Hamiltonian is constructed from the first‐order wave operator to treat several strongly interacting electronic states simultaneously. The wave operator is obtained by solving the generalized Bloch equation within the first‐order interaction space using a multi‐partitioning of the Hamiltonian based on multi‐reference Møller–Plesset second‐order perturbation theory. The corresponding zeroth‐order Hamiltonians are nondiagonal. To reduce the computational effort that arises from the nondiagonal generalized Fock operator, a selection procedure is used that divides the configurations of the first‐order interaction space into two sets based on the strength of the interaction with the reference space. In the weaker interacting set, only the projected diagonal part of the zeroth‐order Hamiltonian is taken into account. The justification of the approach is demonstrated in two examples: the mixing of valence Rydberg states in ethylene, and the avoided crossing of neutral and ionic potential curves in LiF. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

A Lie group method for molecular rovibrational spectra via the broken symmetry of U(r)(2)⊗U(υ)(4)

A special group chain that is appropriate for describing the rovibrational spectra of linear triatomic molecules with respect to both the vibration and the rotation of molecules as harmonic oscillators is given, and the corresponding Hamiltonian is constructed. The eigenvalue expression of the Hamiltonian is similar to the formula commonly used to calculate the rovibrational spectra of linear triatomic molecules. This method eliminates the physical uncertainty brought about by a variety of group chains. The relationships between parameters in the present expression and those in the commonly used expressions are given. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 53–59, 2001     

Ivar Waller 90 years on June 11, 1988

Linkage properties of the diagrammatic representation of the energies obtained in the multireference many-body perturbation calculations with respect to the incompleteness or completeness of the model space are discussed. The case of not completely degenerate model space is considered for which a comparison with the standard single-reference many-body perturbation expansion is possible. The Hose–Kaldor type of graphical representation of the perturbation expansion for the effective Hamiltonian is used in this comparison. It is shown that for an incomplete model space the perturbation expansion is not size-extensive. In this case, for a truncated expansion of the effective Hamiltonian, the energies obtained by diagonalization of the effective Hamiltonian matrix are represented by both linked and unlinked irreducible contributions. The unlinked ones do not appear when the complete model space is used.     

Solution reaction space Hamiltonian based on an electrostatic potential representation of solvent dynamics

Quantum chemical solvation models usually rely on the equilibrium solvation condition and is thus not immediately applicable to the study of nonequilibrium solvation dynamics, particularly those associated with chemical reactions. Here we address this problem by considering an effective Hamiltonian for solution-phase reactions based on an electrostatic potential (ESP) representation of solvent dynamics. In this approach a general ESP field of solvent is employed as collective solvent coordinate, and an effective Hamiltonian is constructed by treating both solute geometry and solvent ESP as dynamical variables. A harmonic bath is then attached onto the ESP variables in order to account for the stochastic nature of solvent dynamics. As an illustration we apply the above method to the proton transfer of a substituted phenol-amine complex in a polar solvent. The effective Hamiltonian is constructed by means of the reference interaction site model self-consistent field method (i.e., a type of quantum chemical solvation model), and a mixed quantum/classical simulation is performed in the space of solute geometry and solvent ESP. The results suggest that important dynamical features of proton transfer in solution can be captured by the present approach, including spontaneous fluctuations of solvent ESP that drives the proton from reactant to product potential wells.     

Global analysis of composite particles

The theory of vibrations of a composite particle when vibrational amplitudes are not constrained to be small according to the Eckart conditions is developed using the methods of differential topology. A global classical Hamiltonian appropriate for this system is given, and for the case of the molecular vibration–rotation problem, it is transformed into a global quantum Hamiltonian operator. It is shown that the zeroth-order term in the global Hamiltonian operator is identical to the Wilson–Howard Hamiltonian; higher-order terms are shown to give successively better approximations to the large amplitude problem. Generalized Eckart conditions are derived for the global classical Hamiltonian; the quantum equivalent of these conditions along with the quantum equivalent of the Eckart conditions are given. The spectrum of the global Hamiltonian operator is discussed and it is shown that the calculation of the vibration–rotation energy states of the system reduces to the same straight-forward procedure, the solution of a secular determinant, as was carried out for the Wilson–Howard Hamiltonian at a later time by Nielsen.     

Exponentially fitted symmetric and symplectic DIRK methods for oscillatory Hamiltonian systems

The order conditions for modified Runge–Kutta methods are derived via the rooted trees. Symmetry and symplecticity conditions and exponential fitting conditions for modified diagonally implicit Runge–Kutta (DIRK) are considered. Three new exponentially fitted symmetric and symplectic diagonally implicit Runge–Kutta (EFSSDIRK) methods of respective second order and fourth order are constructed. Phase properties of the new methods are analyzed. The new EFSSDIRK methods are applied to several Hamiltonian problems and compared to the results obtained by the existing symplectic DIRK methods in the literature.     

Quantum dynamics in macrosystems with several coupled electronic states: hierarchy of effective Hamiltonians

We address the nonadiabatic quantum dynamics of macrosystems with several coupled electronic states, taking into account the possibility of multistate conical intersections. The general situation of an arbitrary number of states and arbitrary number of nuclear degrees of freedom (modes) is considered. The macrosystem is decomposed into a system part carrying a few, strongly coupled modes and an environment, comprising the vast number of remaining modes. By successively transforming the modes of the environment, a hierarchy of effective Hamiltonians for the environment is constructed. Each effective Hamiltonian depends on a reduced number of effective modes, which carry cumulative effects. By considering the system's Hamiltonian along with a few members of the hierarchy, it is shown mathematically by a moment analysis that the quantum dynamics of the entire macrosystem can be numerically exactly computed on a given time scale. The time scale wanted defines the number of effective Hamiltonians to be included. The contribution of the environment to the quantum dynamics of the macrosystem translates into a sequential coupling of effective modes. The wave function of the macrosystem is known in the full space of modes, allowing for the evaluation of observables such as the time-dependent individual excitation along modes of interest as well as spectra and electronic-population dynamics.     

Molecular vibrational energy flow and dilution factors in an anharmonic state space

A fourth-order resonance Hamiltonian is derived from the experimental normal-mode Hamiltonian of SCCl2. The anharmonic vibrational state space constructed from the effective Hamiltonian provides a realistic model for vibrational energy flow from bright states accessible by pulsed laser excitation. We study the experimentally derived distribution PE(sigma) of dilution factors sigma as a function of energy. This distribution characterizes the dynamics in the long-time limit. State space models predict that PE(sigma) should be bimodal, with some states undergoing facile intramolecular vibrational energy redistribution (small sigma), while others at the same total energy remain "protected" (sigma approximately 1). The bimodal distribution is in qualitative agreement with analytical and numerical local density of states models. However, there are fewer states protected from energy flow, and the protected states begin to fragment at higher energy, shifting from sigma approximately 1 to sigma approximately 0.5. We also examine how dilution factors are distributed in the vibrational state space of SCCl2 and how the power law specifying the survival probability of harmonic initial states correlates with the dilution factor distribution of anharmonic initial states.     

Effects of complex-valued electronic wave functions on nuclear motions

Moleculer species and colliding groups of atoms are considered for which the electronic wave functions are complex-valued, having arguments that depend parametrically on the nuclear coordinates. The effective Hamiltonian for nuclear motions in the adiabatic approximation that arises in the present case differs from the ordinary Born–Oppeneheimer Hamiltonian, the latter being obtained when restriction to real-valued electronic functions is made. The asymptotic boundary conditions imposed in collision theory lead to in- and out- states [8], and hence to complex-valued wave functions in the coordinate representation. The study of the influence of electron–molecule scattering on nuclear motions therefore necessitates the use of the new effective Hamiltonian, which leads to results differing from those predicted on the basis of the Born–Oppenheimer operator. It is shown that momentum-dependent potentials occurring in the new Hamiltonian might cause “distortions” to the vibrational patterns of some electron–molecule metastable states. Also, these terms can give rise to non-Born–Oppenheimer resonances when motions in an internuclear coordinate become unbounded. We derive expressions for the relevant level widths and line shapes, showing them to be subject to an isotope effect. Even when real-valued electronic functions may be used, the selections of complex-valued functions in their linear span is still optional. Although exact treatments lead to the same results in both real and complex cases we show how the choice of the argument of the electronic function to be non-zero and dependent on nuclear coordinates may be useful for the application of certain approximation schemes. It is demonstrated that for certain systems a suitable choice of the argument assures convergence when the related Lippmann–Schwinger Equation is iterated. It is also shown that in this way an nth order term in the series expansion of the T matrix [8] for moleculer systems can be made negligibly small.     

Normalized irreducible tensorial matrix expansions applied to an effective sp-type Hamiltonian for the PES of water

Any matrix can be expanded on a basis of SU(2) normalized irreducible tensorial matrices, NITM , defined in terms of 3-j symbols or coupling coefficients of SU (2). The NITM transform under rotations according to Wigner's matrices. If one dimension of an NITM is odd and the other even, the tensor has half-integer rank. A simple NITM basis consists of all NITM having the same numbers of rows and columns as the expanded matrix. A compound NITM basis consists of two or more simple bases, each spanning a corresponding block in the expanded matrix. The choice of NITM basis for expanding an effective Hamiltonian matrix is a crucial step in formulating a model. To illustrate the use of a compound NITM basis, including nonsquare NITM , an effective sp-type overlap-free superposition Hamiltonian is constructed and applied to the photoelectron ionization potential spectrum of water.     

The modified all-valence INDO method with the inclusion of spin–orbit coupling

The all-valence INDO method has been modified for the inclusion of spin–orbit coupling effects. In the method presented, the Hamiltonian includes spin–orbit coupling and the basis set constitutes the singlet and triplet determinental wave functions constructed from molecular orbitals resulting from nonrelativistic calculations. Eigenvectors obtained are later used for the evaluation of transition probabilities among different states. The results presented include lifetimes of different states of organic molecules and transition energies for halogen molecules and they are in a good agreement with experimental results. © 1992 John Wiley & Sons, Inc.     

Canonical transcorrelated theory with projected Slater-type geminals

An effective Hamiltonian perturbed with explicit interelectronic correlation is derived from similarity transformation of Hamiltonian using a unitary operator with Slater-type geminals. The Slater-type geminal is projected onto the excitation (and deexcitation) component as in the F12 theory. Simplification is made by truncating higher-body operators, resulting in a correlated Hamiltonian which is Hermitian and has exactly the same complexity as the original Hamiltonian in the second quantized form. It can thus be easily combined with arbitrary correlation models proposed to date. The present approach constructs a singularity-free Hamiltonian a priori, similarly to the so-called transcorrelated theory, while the use of the canonical transformation assures that the effective Hamiltonian is two-body and Hermite. Our theory is naturally extensible to multireference calculations on the basis of the generalized normal ordering. The construction of the effective Hamiltonian is non-iterative. The numerical assessments demonstrate that the present scheme improves the basis set convergence of the post-mean-field calculations at a similar rate to the explicitly correlated methods proposed by others that couple geminals and conventional excitations.     

Multireference Fock space coupled cluster method in the effective and intermediate Hamiltonian formulation for the (2,0) sector

The effective and intermediate Hamiltonian (IH) multireference coupled cluster (CC) method with singles (S) and doubles (D) within the double electron attached (2,0) sector of the Fock space (FS) is formulated and implemented. The intermediate Hamiltonian realization of the (2,0) FS problem allows to replace the iterative scheme of the FS-CC equations based on the effective Hamiltonian with the diagonalization of the properly constructed matrix. The proposed method, IH-FS-CCSD (2,0), is rigorously size-extensive, easy to code, and numerically very efficient with the results comparable or slightly better than equation-of-motion ones at the CCSDT (T--triples) level. The performance of the method is discussed on the basis of test calculations for potential energy curves of the systems for which double positive ions dissociate into closed shell fragments (e.g., Na(2) dimer). The double electron attachment (DEA) scheme can be also useful in determination of the excitation spectra for difficult cases. The example is a carbon atom which has two electrons out of the closed shell structure. The newly implemented method is also analyzed by plotting potential energy curve for twisted ethylene case as a function of a dihedral angle between two methylene groups. Using DEA scheme one obtains a smooth, cusp free curve.     

On the use of effective core potentials in the calculation of magnetic properties, such as magnetizabilites and magnetic shieldings

State-of-the art effective core potentials (ECPs) that replace electrons of inner atomic cores involve non-local potentials. If such an effective core potential is added to the Hamiltonian of a system in a magnetic field, the resulting Hamiltonian is not gauge invariant. This means, magnetic properties such as magnetisabilities and magnetic shieldings (or magnetic susceptibilities and nuclear magnetic resonance chemical shifts) calculated with different gauge origins are different even for exact solutions of the Schro?dinger equation. It is possible to restore gauge invariance of the Hamiltonian by adding magnetic field dependent terms arising from the effective core potential. Numerical calculations on atomic and diatomic model systems (potassium mono-cation and potassium dimer) clearly demonstrate that the standard effective core potential Hamiltonian violates gauge invariance, and this affects the calculation of magnetisabilities more strongly than the calculation of magnetic shieldings. The modified magnetic field dependent effective core potential Hamiltonian is gauge invariant, and therefore it is the correct starting point for distributed gauge origin methods. The formalism for gauge including atomic orbitals (GIAO) and individual gauge for localized orbitals methods is worked out. ECP GIAO results for the potassium dimer are presented. The new method performs much better than a previous ECP GIAO implementation that did not account for the non-locality of the potential. For magnetic shieldings, deviations are clearly seen, but they amount to few ppm only. For magnetisabilities, our new ECP GIAO implementation is a major improvement, as demonstrated by the comparison of all-electron and ECP results.     

Adiabatic evolution of eigenspectra of model 1-D and 2-D Hamiltonians: Quantum adiabatic switching algorithm in a time-independent Fourier grid Hamiltonian framework

We explore the viability of a time-independent quantum adiabatic switching algorithm in the Fourier grid Hamiltonian (FGH) framework in the presence of degeneracy, avoided crossing, and chaos. The algorithm is simple and cost effective and provides information about the full eigenspectrum of the evolving Hamiltonian. It is shown to be capable of capturing accurately the change in the pattern of level spacing distribution statistics as one switches from a nonchaotic region of parameter values into the chaotic region. The Transition turns out to be less sharp than anticipated. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 133–141, 1998     

Ab initio correlation corrections to the Hartree-Fock quasi band-structure of periodic systems employing Wannier-type orbitals

A size-consistent ab initio formalism to calculate correlation corrections to ionization potentials as well as electron affinities of periodic systems is presented. Our approach is based on a Hartree-Fock scheme which directly yields local orbitals without any a posteriori localization step. The use of local orbitals implies non-zero off-diagonal matrix elements of the Fock operator, which are treated as an additional perturbation and give rise to localization diagrams. Based on the obtained local orbitals, an effective Bloch Hamiltonian is constructed to second order of perturbation theory with all third-order localization diagrams included. In addition, the summation of certain classes of diagrams up to infinite order in the off-diagonal Fock elements as well as the Epstein-Nesbet partitioning of the full Hamiltonian are discussed. The problem of intruder states, frequently encountered in many-body perturbation theory, is dealt with by employing the theory of intermediate Hamiltonians. As model systems we have chosen cyclic periodic structures up to an oligoethylene ring in double-zeta basis; however, the theory presented here straightforwardly carries over to infinite periodic systems. Received: 30 April 1998 / Accepted: 27 July 1998 /  Published online: 7 October 1998     

Correlated one-particle method: numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.     

Exact size consistency of multireference Møller–Plesset perturbation theory

Single‐reference closed‐shell Møller–Plesset perturbation theory is well known for its size consistency, a quality that is essential for consistent comparisons of calculations on molecules of different size. However, it is far from obvious whether this quality can be retained in the multireference case. In this work it is shown that an exactly size consistently generalization to multireference perturbation theory can be constructed. The central result is that the zeroth‐order Hamiltonian should be constructed using separate projection operators for each excitation level, i.e., it should contain no couplings between different excitation levels. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 549–558, 1999