Problems In Quantum Chemistry Research Articles

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.

Compact method for calculating intermolecular interactions

The development of efficient methods for calculating intermolecular interactions (which are responsible for the existence of stable molecular associates, solvation shells, etc.) is a pressing problem of quantum chemistry. We propose a new method for correct calculations of intermolecular interactions, which is based on the solution of SCF equations with fractional occupation numbers. Calculating intermolecular interactions by this method does not require the use of exchange potentials in an explicit form. The method is intended primarily to describe the charge transfer between interacting subsystems. The calculations by this method are compact since the dimensions of matrix problems remain unchanged in the course of the numerical procedure.

Path integral approach to correlation energy

AbstractThe Feynman path integral method is applied to the many‐electron problem of quantum chemistry. We begin with investigating the partition function of the system in question; then, “a classical path of electron” that corresponds to the Hartree–Fock approximation is obtained by minimizing the thermodynamic potential of the system with respect to the electron coordinate. The next‐order approximation is obtained by evaluating the deviation from this classical path, which is approximately written by an easily integrable Gaussian integral. The result is expected to be the random‐phase approximation. As numerical examples, the hydrogen molecule and butadiene are treated. © 1994 John Wiley & Sons, Inc.

Use of approximate integrals in ab initio theory. An application in MP2 energy calculations

We use the resolution of the identify (RI) as convenient way to replace the use of four-index-two-electron integrals with linear combinations of three-index integrals. The method is broadly applicable to a wide range of problems in quantum chemistry. We demonstrate the effectivenes of RI for the calculation of MP2 energies. For the water dimer, agreement within 0.1 kcal/mol is obtained with respect to exact MP2 calculations. The RI-MP2 energies require only about 10% of the time required by conventional MP2.

Comparison of coupled-cluster results with a hybrid of Hartree–Fock and density functional theory

The performance of a hybrid of Hartree–Fock and density functional theory is tested on a set of ‘‘pathological’’ quantum chemistry problems. The predictions of this hybrid model are in qualitative agreement with coupled-cluster results and with experiment. Given the modest computational cost of the procedure, this is an extremely encouraging development.

Analytic evaluation of energy gradients for the singles and doubles coupled cluster method including perturbative triple excitations: Theory and applications to FOOF and Cr2

The analytic energy gradient for the singles and doubles coupled cluster method including a perturbative correction due to triple excitations [CCSD(T)] is formulated and computationally implemented. Encouraged by the recent success in reproducing the experimental equilibrium structure and vibrational frequencies of ozone, the new CCSD(T) gradient method is tested with two other ‘‘difficult’’ quantum chemistry problems: FOOF and Cr2. With the largest basis set employed in this work [triple zeta plus two sets of polarization functions (TZ2Pf)] at the CCSD(T) level of theory, the predictions for the O–O and O–F bond lengths in FOOF are 1.218 and 1.589 Å, respectively. These figures are in good agreement with the experimental values 1.216 and 1.575 Å. Based on CCSD calculations with even larger basis sets, it is concluded that the error of 0.014 Å in the O–F bond length at the TZ2Pf/CCSD(T) level of theory is due to the remaining basis set deficiency. On the other hand, the CCSD(T) prediction for the equilibrium bond length of Cr2 (1.604 Å), obtained with a large (10s8p3d2f1g) basis set capable of achieving the Hartree–Fock limit, is still 0.075 Å shorter than experiment, clearly indicating the importance of higher than connected triple excitations in a single-reference treatment of this particular problem.

Some current problems in quantum chemistry as described by their key words and the International Journal of Quantum Chemistry

In the editorial for the International Journal of Quantum Chemistry, it is said that the journal is devoted to quantum theory and computations in chemistry, condensed matter physics, and biology. In the program for the journal, we have emphasized the significance of the interaction between the formal methods of quantum chemistry, the computational aspects, and the experiments

The nature of chemical structure

We briefly discuss chemical structure as an object of mathematical characterization and list various structural invariants suitable for such a characterization. We restrict our attention to modeling compounds as graphs and examine several diverse structure-property-activity problems as an illustration of different mathematical analyses. Specifically, we consider: (1) the equivalence of an apparent quantum chemical problem as a graph theoretitcal decomposition (the analysis of diamagnetic susceptibility); (2) partial order as a tool for the illustration of regularities in isomeric variation of molecular properties (boiling points in alkanes); (3) ranking of structures as a tool in a search for biologically active substructures, illustrated on mutagenicity of nitrosamines; and (4) construction of search vectors as a tool for finding structures with a prescribed property. We end the discussion by pointing out advantages of mathematical descriptors versus physicochemical properties as descriptors. In conclusion, we focus attention on two problems facing graph theoretical approaches to chemical structure: How to incorporate differences between heteroatoms into molecular graphs, and how to incorporate spatial characteristics of chemical compounds into graph theoretical approaches. Finally, we generalize the traditional graph theoretical approaches, based on graphs and weighted graphs, to physicochemical matrices associated with molecular systems and point out the potential role that structural invariants play in discussions of molecular properties. Within this more general point of view, the quantum chemical computations produce but a fraction of the possible structural invariants that one may consider for a given system.

Die Konstruktion von elektronischen Superiorstrukturen

Using the matrixC of a quantumchemical problem Φ=Cσ, hierarchic classification methods and the connex-tableau-algorithm [20, 21] based on graph theory, superior structure of an electronic system were constructed. These structures are fragments with maximal electronic similarity in relation to the information given byC. It is possible to study the relative stability and the inner structure of the superior fragments. The method is demonstrated for three hydrocarbons with symmetry C1h using results of HMO calculations [24].

Correlations in extended systems: A microscopic multilocal method for describing both local and global properties

We review the basic principles of the various coupled cluster (CC) methods based on an exponential form for the many-body wavefunction, and contrast them with the configuration–interaction (CI) method. Particular emphasis is placed on their applicability to problems in quantum chemistry. We prove that in all cases we can construct an energy functional which variationally determines both the ground–state wavefunction and the dynamic equations of motion for nonstationary states. As a result the equations of motion assume the familiar classical canonical Hamiltonian form in some well-defined (multibody) configuration space. We also thereby construct the expectation-value functional for an arbitrary operator in such a way that the Feynman–Hellmann theorem is preserved at all natural levels of truncation of the appropriate configuration space. We show in detail that only in the case of the recently introduced extended CC method (ECCM) is the expectation-value functional expressed fully in terms of linked (multilocal) amplitudes. The ECCM is thereby capable of describing such global phenomena as shape transitions and other stereochemical properties, and the large-scale behavior of the molecular energy surfaces. We illustrate our methodology on the one-body density matrix, which is now much more easily discussed than by conventional methods in quantum chemistry.

A determinant based full configuration interaction program

The program FCI solves the Full Configuration Interaction (Full CI) problem of quantum chemistry, in which the electronic Schrödinger equation is solved exactly within a given one particle basis set. The Slater determinant based algorithm leads to highly efficient implementation on a vector computer, and has enabled Full CI calculations of dimension more than 10 7 to be performed.

Noncovalent interactions of medium strength. A revised interpretation and examples of its applications

AbstractRecent studies performed in our group on a classical problem of quantum chemistry, with strong implications for theoretical biochemistry and pharmacology, are here summarized. Ab initio descriptions of noncovalent interactions, and in particular H bonds and acid‐base couples, have been reexamined using as novel tools the decomposition of ΔE with the inclusion of CP corrections and a further decomposition of the ΔE components into group contributions. Some results of systematic analyses performed over H‐bonded dimers are reported, supplemented by a successful application of this approach to a problem of noticeable economic importance (the identification of catalysts for the industrial synthesis of tensioactives). A new feature, presented here for the first time, is the extension of the CP‐corrected decomposition of ΔE to bimolecular interactions in solution.

On the use of asymptotic expansions

Divergent asymptotic expansions in quantum chemistry often must be evaluated on Stokes lines, where the form of the expansion changes discontinuously and might appear to be ambiguous. Towards clarifying the use of asymptotic expansions on Stokes lines we discuss by numerical example the Airy function Bi(x) for real, positive x. Two physical problems to which this example is relevant, among others, are the Rayleigh-Schrodinger perturbation theory for the LoSurdo-Stark effect in hydrogen and the JWKB connection-formula problem, for which real series are associated with complex sums. The various roles of partial summation, Pade approximants, and Borel summation are compared. In addition, a derivation is given for an integral that occurs in a simple proof of the Borel summability of asymptotic expansions for the confluent hypergeometric function, which function is fundamental to certain quantum chemistry problems, and which integral is given incorrectly in several standard references.

A parallel architecture and programming language for quantum chemistry

The large gains in computational capability which are required in the future by problems in computational quantum chemistry must come from advances in both parallel architectures and algorithms. The relation between algorithms and architecture is discussed, with examples of non-numerical algorithms for which future architectures should facilitate implementation. The use of timespace complexity trade-offs is discussed. A parallel language extension and architecture targeted towards general numerical and nonnumerical algorithms being developed by Myrias Research Corporation is briefly presented.

Intermediate Hamiltonians as a new class of effective Hamiltonians

The theory of effective Hamiltonians is well established. However, limitations appear in its applicability for many problems in molecular physics and quantum chemistry. The standard effective Hamiltonians may become strongly non-Hermitian when there is a large coupling between the model space, in which they are defined, and the outer space Moreover, in the presence of intruder states, discontinuities appear in the matrix elements of these effective Hamiltonians as a function of the internuclear distances. To solve these difficulties, a new class of effective Hamiltonians (called intermediate Hamiltonians) is presented; only one part of their roots are exact eigen-energies of the full Hamiltonian. The theory of these intermediate Hamiltonians is presented by means of a new wave-operator R which is the analogue of the wave-operator Omega in the theory of effective Hamiltonians. Solutions are obtained by a generalised degenerate perturbation theory (GDPT) and by iterative procedures. Two model systems are numerically solved which demonstrate the good convergence properties of GDPT with respect to standard degenerate perturbation theory (DPT). Continuity of the solutions is also checked in the presence of an intruder state.

Generalized Hellmann–Feynman and curvature theorems in classical and quantum mechanics

AbstractThe generalized Hellmann–Feynman and curvature theorems are derived for the generalized Sturm–Liouville‐type eigenvalue equation which is widely encountered in many branches of physics. In classical mechanics, the Sturm–Liouville eigenvalue equation primarily arises from applying the theory of small oscillations to vibration problems while, in quantum mechanics, the most important example is the time‐independent Schrödinger equation. The generalized theorems, respectively, provide simple useful expressions for the first and second λ derivatives of the eigenvalue where λ is an arbitrary real parameter appearing in the eigenvalue equation. These results, which apply to both classical and quantal systems, include as special cases the well‐known quantum mechanical Hellmann–Feynman and curvature theorems for the time‐independent Schrödinger equation, thus unifying the formalism. This connection between classical and quantum mechanics is of interest per se, and also because classical eigenvalue equations find application in the treatment of some quantum chemical problems. Two derivations of both generalizations are presented, the first proof applying to exact and Rayleigh–Ritz (linear variational) solutions of the eigenvalue problem, and the second, more general one, to arbitrary optimal variational as well as exact solutions. Necessary and sufficient conditions for the validity of the results are discussed. To demonstrate the insight they afford, several applications of the theorems in conjunction with perturbation theory are described.

Molecular orbital resonance theory: A new approach to the treatment of quantum chemical problems

An alternative approach to the treatment of the quantum chemical problems combining both, the MO and VB theory, is proposed. This approach retains the concept of resonance from the VB method, but it treats each particular bond in the MO sense. The method is illustrated with a few examples. Relative stabilities of benzene, pentalene and cyclobutadiene are derived. A Huckel (4m + 2) rule is derived for the annulenes. The charge polarisation in the case of the pentalene molecule is explained. A distortion of the pentalene molecule is considered and it is shown that within this approach the distortion depends on the charge polarisation.

The characteristic polynomial approach to the solution of quantum chemical perturbation problems

An approach to the solution of finite-dimensional quantum chemical perturbation problems based on the ‘characteristic polynomial’ is developed and discussed. Using the successive derivatives of the polynomial, an ordered perturbation sequence is constructed. From this sequence, perturbation series for individual eigenvalues are determined. Applications of this method are illustrated for cases of non-degenerate eigenvalues, two-fold energy degeneracy lifted in first order, andq-fold energy degeneracy. A re-derivation of the well-known Rayleigh-Schrodinger perturbation formulae from their corresponding characteristic polynomial expressions is presented, and an additional illustration is made of the use of the reduced characteristic polynomial by considering a benzene molecule with three adjacent carbon atoms ‘perturbed’ in some way.

A density functional representation of quantum chemistry. II. Local quantum field theories of molecular matter in terms of the charge density operator do not work

AbstractA comprehensive review of the attempts to rephrase molecular quantum mechanics in terms of the particle density operator and the current density or phase density operator is given. All pertinent investigations which have come to our attention suffer from severe mathematical inconsistencies and are not adequate to the few‐body problem of quantum chemistry. The origin of the failure of these attempts is investigated and it is shown that a realization of a local quantum field theory of molecular matter in terms of observables would presuppose the solution of many highly nontrivial mathematical problems.

A density functional representation of quantum chemistry. III. Rigorous realization of the program in lattice space

AbstractA mathematically rigorous reformulation of molecular quantum mechanics in terms of the particle density operator and a canonically conjugated phase field is given. Using a momentum cutoff, it is shown that the usual molecular Hamiltonian can be expressed in terms of the particle density operator and a rigorously defined phase operator. It is shown that this Hamiltonian converges strongly to the cutoff‐free Hamiltonian. In spite of the fact that this Hamiltonian is of second order in the phase operators, all hitherto published expressions are not correct. Unfortunately, the correct formulation destroys the intuitive appeal of using the particle density operator as a coordinate for the many‐body problems of quantum chemistry. Unless somebody provides an essential new and clever idea, we propose to resist the fascination of a local quantum field theory of molecular matter in terms of the particle density operator.