Selection rules in alternant systems

The CI space X_n generated by n electrons moving over 2n spin orbitals is considered. It is shown that the transitions between different eigenstates ψ_i ? X_n of alternant and weakly alternant Hamiltonians are governed by some special selection rules. These selection rules are characteristic to alternant systems, and they do not apply to nonalternant systems. The set of all such selection rules can be easily derived from the splitting theorem. In particular, the selection rules associated with spin independent alternant systems are considered. As an example, the PPP Hamiltonian ?_p describing netural alternant hydrocarbons is treated. In the case of electron dipole transitions between eigenstates ψ_i ? X_n of the Hamiltonian ?_p, the selection rules obtained are in agreement with the selection rules derived previously by Pariser and McLachlan.     

Electronic quantum motions in hydrogen molecule ion

The hydrogen molecule ion is a two‐center force system expressed under the prolate spheroidal coordinates, whose quantum motions and quantum trajectories have never been addressed in the literature before. The momentum operators in this coordinate system are derived for the first time from the Hamilton equations of motion and used to construct the Hamiltonian operator. The resulting Hamiltonian comprises a kinetic energy T and a total potential V_Total consisting of the Coulomb potential and a quantum potential. It is shown that the participation of the quantum potential and the accompanied quantum forces in the force interaction within H₂⁺ is essential to develop an electronic motion consistent with the prediction of the probability density function |Ψ|². The motion of the electron in H₂⁺ can be either described by the Hamilton equations derived from the Hamiltonian H = T_K + V_Total or by the Lagrange equations derived from the Lagrangian H = T_K ? V_Total. Solving the equations of motion with different initial positions, we show that the solutions yield an assembly of electronic quantum trajectories whose distribution and concentration reconstruct the σ and π molecular orbitals in H₂⁺. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Study of the kinetic energy densities of electrons as applied to quantum dots in a magnetic field

There are three expressions for the kinetic energy density t( r ) expressed in terms of its quantal source, the single-particle density matrix: t _A( r ) , the integrand of the kinetic energy expectation value; t _B( r ) , the trace of the kinetic energy tensor; t _C( r ) , a virial form in terms of the ‘classical’ kinetic field. These kinetic energy densities are studied by application to ‘artificial atoms‘ or quantum dots in a magnetic field in a ground and excited singlet state. A comparison with the densities for natural atoms and molecules in their ground state is made. The near nucleus structure of these densities for natural atoms is explained. We suggest that in theoretical frameworks which employ the kinetic energy density such as molecular fragmentation, density functional theory, and information-entropic theories, one use all three expressions on application to quantum dots, and the virial expression for natural atoms and molecules. New physics could thereby be gleaned.     

Semiempirical Hamiltonians for spatially confined π-electron systems

A study of π-electron systems confined by impenetrable surfaces is presented. The study results in a nonempirical-based approach to obtain confinement-adapted semiempirical π-Hamiltonians including repulsive terms (PPP or Hubbard). The impenetrable surface confinement of a physical system involves changes in the boundary conditions that the eigenvectors of its differential Hamiltonian operator have to fulfill, while the Hamiltonian itself remains unchanged. However, if this Hamiltonian is written in second quantization language, then confinement only involves changes of the Hamiltonian scalar factors (integrals). Semiempirical Hamiltonian integrals are replaced by parameters; therefore, confinement involves only changes of these parameters. It is shown that confinement changes Coulomb (α_i) and exchange (β_ij), while repulsion (γ_ij) parameters remain unaffected. Next, the influence of confinement upon the electron correlation of (i) π-electron molecular systems, (ii) atoms, and (iii) an electron gas is discussed. The behaviour of the correlation energy vs. the confinement size is found to be different for each type of system. A neat explanation of this variety is given in terms of the Coulomb attractive fields of the systems. Some chemical confinement effects such as an increase in the reactivity of π-electron systems is also outlined. © 1996 John Wiley & Sons, Inc.     

Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Gel formation with multiple interunit junctions. I—Molecules carrying different functional groups

Number‐ and weight‐average molecular weight of condensation polymers formed by primary molecules carrying different species of functional groups {A_i} (i = 1, 2, …, s) are derived by cascade theory. These functional groups are allowed to form multiple junctions of variable multiplicity k. The gel point condition is found to be given by ∑ w_i/|μ_w,i + 1/∑ f_i ? 1 = 0, where f_i is the number of A_i groups specified by the index i on a primary molecule, w_i ≡ f_i/∑ f_i the number fraction of the species i it carries, and |μ_w,i the weight average multiplicity of the junctions formed by the groups A_i. The explicit form of the molecular weight distribution function is found for the simplest case of two components. Possible application to thermoreversible gelation is suggested. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2405–2412, 2003     

Polymer drying. VIII. Molecular interpretations of desorption from polystyrene–liquid systems

The results obtained in time studies that monitored evaporation from liquid-saturated poly(styrene-co-divinylbenzene) to virtual dryness at temperatures ranging from 20 to 80°C confirm those reported earlier for multireplicated time studies at 23 ± 1°C; i.e., when the residual composition (α_t, in sorbed molecules per phenyl group) attains the glassy state composition, the value of α_t thereafter is given by a linear combination of no more than six exponential decay functions. The logarithms of the rate constants (k_i) for decay of these populations at a given temperature decreased linearly with i, the population identification number in the order of decreasing decay rate. The Arrhenius activation energy (ΔE_i) for increase in k_i with temperature was characteristic of the sorbed species, but it was independent of i. The logarithms of the frequency factors (A_i) decreased linearly with i, the slope of which was numerically equal to that observed for the corresponding k_i relationships, signifying that the stepwise decrease of the latter at a given temperature is attributable primarily to a corresponding incremental decrease in entropy. It is suggested that this reflects discrete changes in the molecular structure of polymeric inclusion complexes formed during the transition from the rubbery to the glassy state, as discussed in the text. © 1993 John Wiley & Sons, Inc.     

Representations of Shavitt Graphs Within the Graphical Unitary Group Approach

The Shavitt graph is a visual representation of a distinct row table (DRT) within the graphical unitary group approach. The DRT is a compact representation of the entire configuration state function expansion space within a molecular electronic structure calculation. Each node of the graph is associated with an integer triple (a _k,b _k,c _k). These integers may be mapped to other quantum numbers, including the number of orbitals, number of electrons, and spin quantum number, and used to display Shavitt graphs in various ways that emphasize different aspects of the expansion space or that reveal different aspects of computed wave functions. The features of several graph density plots are discussed, including electron–hole symmetries and the bonding–antibonding wave function character. © 2019 Wiley Periodicals, Inc.     

Nonadiabatic effects in the nuclear probability and flux densities through the fractional Schrödinger equation

Nonadiabatic effects in the nuclear dynamics of the H₂⁺ molecular ion, initiated by ionization of the H₂ molecule, is studied by means of the probability and flux distribution functions arising from the space fractional Schrödinger equation. In order to solve the fractional Schrödinger eigenvalue equation, it is shown that the quantum Riesz fractional derivative operator fulfills the usual properties of the quantum momentum operator acting on the bra and ket vectors of the abstract Hilbert space. Then, the fractional Fourier grid Hamiltonian method is implemented and applied to molecular vibrations. The eigenenergies and eigenfunctions of the fractional Schrödinger equation describing the vibrational motion of the H₂⁺ and D₂⁺ molecules are analyzed. In particular, it is shown that the position-momentum Heisenberg's uncertainty relationship holds independently of the fractional Schrödinger equation. Finally, the probability and flux distributions are presented, demonstrating the applicability of the fractional Schrödinger equation for taking into account nonadiabatic effects.     

On the formulation of spin-free quantum chemistry

A set of spin-free functions ?_i(r),i = 1 …? f, is obtained which form the basis of spin-free quantum chemistry. The ?_i(r) show a one-to-one correspondence to antisymmetric space-spin functions Ψ_i(r, σ) with spin functions constructed according to Löwdin's projector operator method.     

Free energy functional of the Ising-like model of spin crossover

Abstract  

The model of spin crossover based on the Ising-like Hamiltonian (IHM) has been analysed by deriving the functional of free energy from the mean-field solutions of this Hamiltonian. The contribution of the configurational entropy was found to be identical to that in the functional of the molecular statistical model (MSM) of spin crossover. However, the polynomial expansion over composition (x _B) and degree of order (s _B) in these functionals differ fundamentally due to different ways of accounting for the effects of molecular interactions. It was found that IHM takes into account next-to-nearest neighbour interactions by introducing affinities of sublattices towards molecules of given kinds. This yields a term proportional to the first power of the degree of order in the functional of IHM, whereas the MSM free energy is only proportional to s _B². The choice of formal independent variables does not affect the results of simulations of transition curves provided the functional remains unaltered. This provides for more flexibility in numerical simulations of transition curves.     

Non-local relation between kinetic and exchange energy densities in Hartree–Fock theory

A non-local generalization K( r, r' ) of the kinetic energy t( r ) such that t( r ) = ∫K( r, r' ) dr' is defined using the idempotency property of the Hartree–Fock first-order density matrix. This is, in turn, related by means of an explicit differential equation to the non-local exchange energy density X( r, r' ). The relationship is illustrated for a couple of examples: with the Fermi-hole in a uniform electron gas, of importance in the local density version of density functional theory, and with inhomogeneous electron systems.     

On the long way from the Coulombic Hamiltonian to the intermolecular energy surfaces: Concluding remarks at the tromsø conference,June 20–22, 1989

After a brief review of the main results given at the conference, the general properties of the Coulombic Hamiltonian for a system of electrons moving in a framework of moving atomic nuclei—considered as point charges—are discussed. Since this Hamiltonian is invariant under translations and rotations, the total momentum and the total angular momentum are constants of motion, which means that it is possible to separate the motion of the center of mass and the rotation of the system as a whole. Even if these separations are simple in principle, they lead to a mixing of the electronic and nuclear coordinates that complicates the transformed Hamiltonian. The general features of this Hamiltonian are discussed both in pure quantum mechanics and general quantum theory dealing with wave functions Ψ respective density matrices ρ or system operators T. The principles of the latter are derived from five simple axioms, and it is shown that pure quantum mechanics is a special case of the general theory and that the analogy between these two approaches is essential for the “economy of thinking.” It is indicated that the general theory of the shape and topology of the energy surface 〈H〉 = TrHΓ and its critical points, as a function of the system operator Γ involving both electronic and nuclear coordinates, is a very difficult mathematical problem and that calculation of this surface even for simple molecular systems represents a formidable computational problem, which has to be solved in order to be able to understand the nature of chemical reactions from first principles.     

Time scales and other problems in linking simulations of simple chemical systems to more complex ones

We shall start with very small systems like H₂ and H₃, computed with very accurate methods (Hylleraas–CI ) or atomic systems up to Zn with accurate methods (CI ), then move to more complex ones, like C₆₀, but now with somewhat less accurate methods, specifically Hartree–Fock with density functionals, the latter for the correlation energy but not for the exchange energy. For even more complex tasks like geometry optimization of C₆₀, we have resorted to even simpler and parametrized methods, like local density functionals. Then, we could use quantum mechanics either to provide interaction potentials for classical molecular dynamics or to directly solve dynamical systems, in a quantum molecular dynamics approximation. Having demonstrated that we can use the computational output from small systems as input to larger ones, we discuss in detail a new model for liquid water, which is borne out entirely from ab initio methods and nicely links spectroscopic, thermodynamics, and other physicochemical data. Concerning time scales, we use classical molecular dynamics to determine friction coefficients, and with these we perform stochastic dynamic simulations. The use of simulation results from smaller systems to provide inputs for larger system simulations is the “global simulation” approach, which, today, with the easily available computers, is becoming more and more feasible. Projections on simulations in the 1996–1998 period are discussed, new computational areas are outlined, and a N⁴ complexity algorithm is compared to density functional approaches. © 1993 John Wiley & Sons, Inc.     

Six-dimensional tunneling dynamics of internal rotation in hydrogen peroxide molecule and its isotopomers

The six-dimensional torsion-vibration Hamiltonian of the H₂O₂ molecule and its H/D- and ¹⁸O/¹⁶O-isotopomers is derived. The Hamiltonian includes the kinetic energy operator, which depends on the tunneling coordinate, and the potential energy surface represented as a quartic polynomial with respect to the small-amplitude transverse coordinates. Parameters of the Hamiltonian were obtained from DFT calculations of the equilibrium geometries, eigenvectors, and eigenfrequencies of normal vibrations at the stationary points corresponding to the ground state and both the cis- and trans-transition states, carried out with the B3LYP density functional and 6-311+G(2d,p) basis set. The quantum dynamics problem is solved using the perturbative instanton approach generalized for the excited states situated above the barrier top. Vibration-tunneling spectra are calculated for the ground state and low-lying excited states with energies below 2000 cm^–1. Strong kinematic and squeezed potential couplings between the large-amplitude torsional motion and bending modes are shown to be responsible for the vibration-assisted tunneling and for the dependence of tunneling splittings on the quantum numbers of small-amplitude transverse vibrations. Mode-specific isotope effects are predicted.     

Molecular interaction studies in hydrogen bonded polar binary mixtures

The molecular interactions between the polar systems of propan-1-ol (1PN) with alkyl benzoates (methyl benzoate and ethyl benzoate) for various mole fractions at different temperatures are studied by determining the dielectric permittivity in radio, microwave and optic frequency regions, respectively. Dipole moment, excess dipole moment, excess Helmholtz free energy, excess permittivity, relaxation time, excess inverse relaxation time and excess thermodynamical values are calculated using experimental data. Hamiltonian quantum mechanical calculations are performed on both pure and equimolar binary systems of 1PN with alkyl benzoates for the measurement of dipole moment from the ab initio Hartree–Fock and density functional theory (B3LYP) methods with 6-31?+?G* and 6-311?+?G** basis sets using Spartan 08 modelling software and these theoretical values are in good agreement with the experimental values.     

Polymerization of styrene with VOCl3–aluminum alkyls

The polymerization of styrene with VOCl₃ in combination with AlEt₃ and with Al(i-Bu)₃ in n-hexane at 40°C. has been investigated. The rate of polymerization was found to be second order with respect to monomer in both systems. With respect to catalyst the rate of polymerization was first order for VOCl₃–AlEt₃ and second order for VOCl₃-Al(i-Bu)₃ systems. The activation energies for VOCl₃–AlEt₃ and VOCl₃–Al(i-Bu)₃ systems were 7.37 and 11.25 kcal./mole, respectively. The molecular weight of polystyrene in the AlEt₃ system was considerably higher than that in the Al(i-Bu)₃ system. The valence of vanadium obtained by a potentiometric method showed that the catalyst sites in the AlEt₃ system are different in nature from those in the Al(i-Bu)₃ system. The effect of diethylzinc as a chain-transfer agent in the AlEt₃ system was also studied.     

分子长程作用导致的电子态转移半径典量子理论

以半经典量子力学方法发展了一个广义的理论,用来描述分子间长程作用下碰撞导致的电子态跃迁过程。运用分子间特别是线性分子间的一级静电Hamiltonian推导碰撞导致的电子态跃迁矩阵元的表达式,分别考虑了共振和近共振两种能量转移的情形,获得了对应于电子和分子转动运动的选律。并将理论表达式用于碰撞导致的碘分子E带E⁺_{0g和D0u^{+离子对态的数值计算。}}     

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

A 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.