Generalized quantum mechanical two-centre problems

If χ _i (χ _k ) is an exact generalized diatomic orbital (solution of Eq. (1) of text), a sequence of functions χ _i ^(N) converging to χ _i may be constructed so that matrix elements of frequently occurring operators between χ _i ^(N) and χ _k ^(N) may be computed without any numerical integration. Exact expectation values are given for kinetic and potential energy, dipole moment, θ ²=x ²+y ², and quadrupole moment 3z ²?r ², for various ratios of nuclear charges Z ₁,Z ₂ and for several distances R. Special subjects discussed in terms of computed expectation values are:

1. R-dependence of the contributions to total energy of HeH²⁺ in state 2pσ and of LiH³⁺ in state 3dσ
2. RZ-and λ-dependence of dipole and quadrupole moment functions in state 1sσ
3. Some properties of those generalized diatomic orbitals which approach, for R going to 0, Slater-type atomic functions.

     

Generalized quantum mechanical two-centre problems

Correlation diagrams for the lowest electronic states of the systems (LiH)³⁺, (HeH)⁺⁺, (LiHe)⁴⁺, H ₂ ⁺ , (He, –1), and (H, –1)^– (finite dipole with one electron) have been computed exactly and are discussed with special regard to the non-crossing rule and to the asymptotic behaviour of generalized diatomic orbitals.

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Zusammenfassung Korrelationsdiagramme für die tiefsten elektronischen Zustände der Systeme (LiH)³⁺, (HeH)⁺⁺, (LiHe)⁴⁺, H ₂ ⁺ , (He, –1) und (H, –1)^– (endlicher Dipol mit einem Elektron) wurden exakt berechnet und werden im Hinblick auf die Nichtüberschneidungsregel und auf das asymptotische Verhalten verallgemeinerter zweiatomiger Bahnfunktionen diskutiert.

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Résumé On a calculé exactement les diagrammes de corrélation pour les plus bas états électroniques des systèmes (LiH)³⁺, (HeH)⁺⁺, (LiHe)⁴⁺, H ₂ ⁺ , (He, –1) et (H, –1) (dipole fini à 1 électron). La discussion porte plus spécialement sur la loi de non-intersection et sur le comportement asymptotique des orbitales diatomiques généralisées.

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Dedicated to Prof. Dr. Hermann Dänzer on occasion of his 65 th birthday on October 21 st.

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The authors express their thanks to the Deutsche Forschungsgemeinschaft for financing computer time on the IBM 7094 and the TR 440 of the Deutsches Rechenzentrum Darmstadt and the CD 3300 of the University of Mainz.     

Generalized quantum mechanical two-centre problems. IV. On the accuracy of simple LCAO approximations for H 2 + states and the Eckart criterion

If x denotes an exact solution of the quantum mechanical two centre Coulomb problem, we optimize a normalized LCAO approximation by making the overlap S = (x¦) a maximum. In this context we study how a weight factor (r _a r _b )^–1 in the definition of the inner product changes the approximation and the expectation value of electronic energy. Finally we compare the lower bound given by the Eckart criterion with the exact overlap. Results are reported for H ₂ ⁺ states 1s_g and 2p_u.Dedicated to Professor Hermann Hartmann on occasion of his 70th birthday on May 4th, 1984     

Extended universal correlation diagrams for the quantum mechanical one-electron two-centre problem

We have recomputed exact correlation diagrams for the energy of excited states; the system consists of an electron (or a different negatively charged particle) moving in the field of two fixed point chargesZ _a ,Z _b . The ratiosq=Z _a/Z_b taken into consideration here were 1, 2, and 3. In the caseq=3, three avoided crossings of energy curves belonging to states of equal symmetry type are contained in the diagram.     

Generalized monotonically convergent algorithms for solving quantum optimal control problems

A wide range of cost functionals that describe the criteria for designing optimal pulses can be reduced to two basic functionals by the introduction of product spaces. We extend previous monotonically convergent algorithms to solve the generalized pulse design equations derived from those basic functionals. The new algorithms are proved to exhibit monotonic convergence. Numerical tests are implemented in four-level model systems employing stationary and/or nonstationary targets in the absence and/or presence of relaxation. Trajectory plots that conveniently present the global nature of the convergence behavior show that slow convergence may often be attributed to "trapping" and that relaxation processes may remove such unfavorable behavior.     

A quantum mechanical curvature theorem

A general quantum mechanical curvature theorem of the form d²ϵ/dχ² ⩽ 〈ψ¦d²H/d벦ψ〉 is established by means of perturbational-variational theory; here, ψ and ϵ are the exact eigenfunctions and eigenvalues of the Schrödinger time-independent equation, H is the Hamilton operator, and λ is any real parameter occurring in H. The theorem is also established for arbitrary optimum variational solutions to the Schrödinger equation. Several applications of the curvature theorem are discussed in conjunction with perturbation theory and the stability of approximate wave functions.     

Hermitian operators for two-centre wavefunctions

Semi-analytical solutions of the Schrödinger equation for a particle moving in the electrostatic field of two other particles a fixed distance apart, are derived in such a way that the resulting matrix eigenvalue equations contain real symmetric band matrices. Numerical techniques appropriate to the solution of the two simultaneous matrix eigenvalue equations are described; in particular the bisection method is used to determine precisely the significant truncation order of the matrices for a given numerical precision.     

Generalized hybrid orbital for the treatment of boundary atoms in combined quantum mechanical and molecular mechanical calculations using the semiempirical parameterized model 3 method

The application of combined quantum mechanical (QM) and molecular mechanical methods to large molecular systems requires an adequate treatment of the boundary between the two approaches. In this article, we extend the generalized hybrid orbital (GHO) method to the semiempirical parameterized model 3 (PM3) Hamiltonian combined with the CHARMM force field. The GHO method makes use of four hybrid orbitals, one of which is included in the QM region in self-consistent field optimization and three are treated as auxiliary orbitals that do not participate in the QM optimization, but they provide an effective electric field for interactions. An important feature of the GHO method is that the semiempirical parameters for the boundary atom are transferable, and these parameters have been developed for a carbon boundary atom consistent with the PM3 model. The combined GHO-PM3/CHARMM model has been tested on molecular geometry and proton affinity for a series of organic compounds.Acknowledgement We thank the National Institutes of Health for support of this research.Contribution to the Jacopo Tomasi Honorary Issue     

Spatially confined simple quantum mechanical systems

A study of two simple quantum mechanical system, i.e., harmonic oscillator (HO ) and hydrogen atom (H ) trapped inside the wells is performed. The influence of the spatial confinement on the energy spectra of both systems is investigated. An analytically solvable model of the well potential is presented in the case of the HO . A central and noncentral localization of HO in the well is discussed. The results are used to explain some reported anomalous experimental data on the porphine IR spectrum. The case of hydrogen atom is studied by solving the problem of the electron in the potential V = Z/R cot r/R. An exact formula for the energy is derived for s states. For others, analytically nonsolvable symmetries, a formula with correct asymptotic behavior is proposed. The results are compared with some numerical calculations for spherical rectangular well and an extension to many-electron atoms is also presented. The effect of breaking degeneracy and of ordering energies of H in a different way than in the case of many-electron systems is broadly discussed. © 1994 John Wiley & Sons, Inc.     

Specific quantum mechanical/molecular mechanical capping-potentials for biomolecular functional groups

We present a series of capping-potentials designed as link atoms to saturate dangling bonds at the quantum/classical interface within density functional theory-based hybrid QM/MM calculations. We aim at imitating the properties of different carbon-carbon bonds by means of monovalent analytic pseudopotentials. These effective potentials are optimized such that the perturbations of the quantum electronic density are minimized. This optimization is based on a stochastic scheme, which helps to avoid local minima trapping. For a series of common biomolecular groups, we find capping-potentials that outperform the more common hydrogen-capping in view of structural and spectroscopic properties. To demonstrate the transferability to complex systems, we also benchmark our potentials with a hydrogen-bonded dimer, yielding systematic improvements in structural and spectroscopic parameters.     

Ewald mesh method for quantum mechanical calculations

The Fourier transform Coulomb (FTC) method has been shown to be effective for the fast and accurate calculation of long-range Coulomb interactions between diffuse (low-energy cutoff) densities in quantum mechanical (QM) systems. In this work, we split the potential of a compact (high-energy cutoff) density into short-range and long-range components, similarly to how point charges are handled in the Ewald mesh methods in molecular mechanics simulations. With this linear scaling QM Ewald mesh method, the long-range potential of compact densities can be represented on the same grid as the diffuse densities that are treated by the FTC method. The new method is accurate and significantly reduces the amount of computational time on short-range interactions, especially when it is compared to the continuous fast multipole method.     

Hofmeister series: The quantum mechanical viewpoint

It is suggested that electromagnetic quantum vacuum fluctuations are at the very deep root of the so-called “specific ions effects” in concentrated solutions or in living cells. A many-body quantum-mechanical frame of thinking is proposed based on the concept of quantum coherence taking into account explicitly density and excitation frequencies of molecules and/or ionic species. It is also proposed that Hofmeister phenomena could have a natural explanation in the harmonic relationships between sets of characteristic frequencies ruled by quantum mechanical laws. It then follows that physical chemistry of concentrated media and biology should be ruled more by a quantum “symphony” between indistinguishable constituents rather than localized two-body electrical interactions between molecular or ionic species.     

A quantum mechanical test of diophantine methods

A blend of Haselgrove's method and the biased selection method for evaluating multidimensional integrals was tested. The results were mixed. The error estimate varied from being proportional to 1/N when N was less than ca. 60,000 to being proportional to 1/√N when N was greater than 60,000. Also, for N greater than 60,000, the error estimate was one-half the error estimate given by biased selection alone. These numbers should be compared with the 10,000 points used to find an optimum set of Haselgrove's parameters. It is reasonable to expect that if 100,000 points were used in the optimization of Haselgrove's parameters that the above results would be found with 60,000 replaced by 600,000.     

Mulliken population analysis and quantum mechanical probability

It is shown that, under certain conditions, the use of Mulliken gross atom populations for the analysis of molecular electronic wavefunctions is not without a basis in the rigorous probabilistic interpretation of quantum mechanics,     

Linear-scaling quantum mechanical methods for excited states

The poor scaling of many existing quantum mechanical methods with respect to the system size hinders their applications to large systems. In this tutorial review, we focus on latest research on linear-scaling or O(N) quantum mechanical methods for excited states. Based on the locality of quantum mechanical systems, O(N) quantum mechanical methods for excited states are comprised of two categories, the time-domain and frequency-domain methods. The former solves the dynamics of the electronic systems in real time while the latter involves direct evaluation of electronic response in the frequency-domain. The localized density matrix (LDM) method is the first and most mature linear-scaling quantum mechanical method for excited states. It has been implemented in time- and frequency-domains. The O(N) time-domain methods also include the approach that solves the time-dependent Kohn-Sham (TDKS) equation using the non-orthogonal localized molecular orbitals (NOLMOs). Besides the frequency-domain LDM method, other O(N) frequency-domain methods have been proposed and implemented at the first-principles level. Except one-dimensional or quasi-one-dimensional systems, the O(N) frequency-domain methods are often not applicable to resonant responses because of the convergence problem. For linear response, the most efficient O(N) first-principles method is found to be the LDM method with Chebyshev expansion for time integration. For off-resonant response (including nonlinear properties) at a specific frequency, the frequency-domain methods with iterative solvers are quite efficient and thus practical. For nonlinear response, both on-resonance and off-resonance, the time-domain methods can be used, however, as the time-domain first-principles methods are quite expensive, time-domain O(N) semi-empirical methods are often the practical choice. Compared to the O(N) frequency-domain methods, the O(N) time-domain methods for excited states are much more mature and numerically stable, and have been applied widely to investigate the dynamics of complex molecular systems.     

Molecular model with quantum mechanical bonding information

The molecular structure can be defined quantum mechanically thanks to the theory of atoms in molecules. Here, we report a new molecular model that reflects quantum mechanical properties of the chemical bonds. This graphical representation of molecules is based on the topology of the electron density at the critical points. The eigenvalues of the Hessian are used for depicting the critical points three-dimensionally. The bond path linking two atoms has a thickness that is proportional to the electron density at the bond critical point. The nuclei are represented according to the experimentally determined atomic radii. The resulting molecular structures are similar to the traditional ball and stick ones, with the difference that in this model each object included in the plot provides topological information about the atoms and bonding interactions. As a result, the character and intensity of any given interatomic interaction can be identified by visual inspection, including the noncovalent ones. Because similar bonding interactions have similar plots, this tool permits the visualization of chemical bond transferability, revealing the presence of functional groups in large molecules.     

Improved pseudobonds for combined ab initio quantum mechanical/molecular mechanical methods

The pseudobond approach offers a smooth connection at the quantum mechanical/molecular mechanical interface which passes through covalent bonds. It replaces the boundary atom of the environment part with a seven-valence-electron atom to form a pseudobond with the boundary atom of the active part [Y. Zhang, T. S. Lee, and W. Yang, J. Chem. Phys. 110, 46 (1999)]. In its original formulation, the seven-valence-electron boundary atom has the basis set of fluorine and a parametrized effective core potential. Up to now, only the Cps(sp3)-C(sp3) pseudobond has been successfully developed; thus in the case of proteins, it can only be used to cut the protein side chains. Here we employ a different formulation to construct this seven-valence-electron boundary atom, which has its own basis set as well as the effective core potential. We have not only further improved Cps(sp3)-C(sp3) pseudobond, but also developed Cps(sp3)-C(sp2,carbonyl) and Cps(sp3)-N(sp3) pseudobonds for the cutting of protein backbones and nucleic acid bases. The basis set and effective core potential for the seven-valence-electron boundary atom are independent of the molecular mechanical force field. Although the parametrization is performed with density functional calculations using hybrid B3LYP exchange-correlation functional, it is found that the same set of parameters is also applicable to Hartree-Fock and MP2 methods, as well as DFT calculations with other exchange-correlation functionals. Tests on a series of molecules yield very good structural, electronic, and energetic results in comparison with the corresponding full ab initio quantum mechanical calculations.