Valence Bond Wave Functions Research Articles

On the interpretation of valence bond wavefunctions

Valence bond wavefunctions are naturally geared to the chemist's idea of chemical bonding. In a structure one may distinguish different electron pair bonds and possible radical character. A structure may correspond to a covalent bond, where all electrons are equally divided over the atoms, or may describe an excess charge on a discrete part of the molecule, which indicates ionic bond character. From the weights of the structure in a variationally optimised multi-structure valence bond wavefunction one may derive the importance of the different bonding types. The individual structures could be considered to represent the different physical situations. We explore this concept for simple diatomic molecules and for polyatomics, and we discuss the relation to Lewis structures. We show that the assumption of individual properties for the individual structures leads to inconsistencies.

Optimized effective potential from a correlated wave function: Optimized effective potential-generalized valence bond (OEP-GVB)

The optimized effective potential (OEP) method allows orbital-dependent functionals to be used in density functional theory. Traditionally the orbital-dependent functional of interest has been the Hartree-Fock energy, leading to exact exchange density functional theory. Here we present results that use a generalized valence-bond (GVB) wave function, a multiconfigurational wave function that includes static correlation and dissociates to the proper limits. We demonstrate the effectiveness of the OEP-GVB method by showing the dissociation of H2 and the excitation spectrum of He.

Coupled valence bond theory

In this Letter, the formulation and implementation of a parallel response property code for non-orthogonal, valence bond wavefunctions are described. Test calculations on benzene and cyclobutadiene show that the polarisability and magnetisability tensors obtained using valence bond theory are comparable to those obtained at the full π-CASSCF level of theory. The properties are a weighted average of those of its individual resonance structures. Resonance has no relevance for the polarisability and magnetisability.

One-band Hubbard model with hopping asymmetry and the effective theory at finiteU: Phase diagram and metal-insulator transition

We study the one-band Hubbard model at half filling with hopping asymmetry and its effective model at finite but large $U$ up to the second order of ${t}_{\text{mix}}∕U$. Two variational wave functions, the resonating valence bond (RVB) wave function and antiferromagnetic (AF) RVB coexisted wave function, are studied by variational Monte Carlo method on $L\ifmmode\times\else\texttimes\fi{}L$ square lattices up to $L=12$. Based on these two wave functions, the phase diagrams for both models are presented. For the Hubbard model, we find that there is a metal-insulator transition when the hopping parameter ${t}_{\text{mix}}$ which changes the local double occupant vanishes while only a metal-insulator crossover is explored for any finite ${t}_{\text{mix}}$. For the effective model in which the perturbation expansion is up to the second order of ${t}_{\text{mix}}∕U$, a clear metal-insulator transition can be identified for both variational wave functions and the phase diagram can be drawn accordingly. In both models, we find that the systems are dominated by AF-RVB wave function when $U$ is large while the RVB wave function is favored when $U$ is small.

Bosonic resonating valence bond wave function for doped Mott insulators

We propose a class of ground states for doped Mott insulators in the electron second quantization representation. They are obtained from a bosonic resonating valence bond (RVB) theory of the $t\text{\penalty1000-\hskip0pt}J$ model. At half filling, the ground state describes spin correlations of the $S=1∕2$ Heisenberg model very accurately. Its spin degrees of freedom are characterized by RVB pairing of spins, the size of which decreases continuously as holes are doped into the system. Charge degrees of freedom emerge upon doping and are described by twisted holes in the RVB background. We show that the twisted holes exhibit an off diagonal long range order (ODLRO) in the pseudogap ground state, which has a finite pairing amplitude, but is short of phase coherence. Unpaired spins in such a pseudogap ground state behave as free vortices, preventing superconducting phase coherence. The existence of nodal quasiparticles is also ensured by such a hidden ODLRO in the ground state, which is non-Fermi liquidlike in the absence of superconducting phase coherence. Two distinct types of spin excitations can also be constructed. The superconducting instability of the pseudogap ground state is discussed and a $d$-wave superconducting ground state is obtained. This class of pseudogap and superconducting ground states unifies antiferromagnetism, pseudogap, superconductivity, and Mott physics into a new state of matter.

The Electronic Structure of Inorganic Benzenes: Valence Bond and Ring-Current Descriptions

Valence bond (VB) theory and ring-current maps have been used to study the electronic structure of inorganic benzene analogues X(6)H(6) (X = C (1), Si (2)), X(6) (X = N (3), P (4)), X(3)Y(3)H(6) (X,Y = B,N (5), B,P (6), Al,N (7), Al,P (8)), and B(3)Y(3)H(3) (Y = O (9), S (10)). It is shown that the homonuclear compounds possess benzene-like character, with resonance between two Kekulé-like structures and induced diatropic ring currents. Heteronuclear compounds typically show localization of the lone pairs on the electronegative atoms; Kekulé-like structures do not contribute. Of the heteronuclear compounds, only B(3)P(3)H(6) (6) has some benzene-like features with a significant contribution of two Kekulé-like structures to its VB wave function, an appreciable resonance energy, and a discernible diatropic ring current in planar geometry. However, relaxation of 6 to the optimal nonplanar chair conformation is accompanied by the onset of localization of the ring current.

Electron density, exchange-correlation density, and bond characterization from the perspective of the valence-bond theory. I. Two simple analytical cases

In this work, using a valence-bond wave function we obtain analytical expressions for the first- and second-order reduced density matrices of two simple, but quite representative, cases of diatomic molecular systems, namely, H2 and LiH. A detailed study of their exchange-correlation density is performed for both equilibrium and nonequilibrium internuclear distances, discriminating the parallel- and antiparallel-spin contributions. The results show that the behavior of the exchange-correlation density clearly changes with the character of the bond, making it possible to obtain a good deal of information regarding the type of the bond interaction.

Resonating valence bond wave function: from lattice models to realistic systems

Although mean field theories have been very successful to predict a wide range of properties for solids, the discovery of high temperature superconductivity in cuprates supported the idea that strongly correlated materials cannot be qualitatively described by a mean field approach. After the original proposal by Anderson [Science 235 (1987) 1196], there is now a large amount of numerical evidence that the simple but general resonating valence bond (RVB) wave function contains just those ingredients missing in uncorrelated theories, so that the main features of electron correlation can be captured by the variational RVB approach. Strongly correlated antiferromagnetic (AFM) systems, like Cs 2CuCl 4, displaying unconventional features of spin fractionalization, are also understood within this variational scheme. From the computational point of view the remarkable feature of this approach is that several resonating valence bonds can be dealt simultaneously with a single determinant, at a computational cost growing with the number of electrons similarly to more conventional methods, such as Hartree–Fock or Density Functional Theory. Recently several molecules have been studied by using the RVB wave function; we have always obtained total energies, bonding lengths and binding energies comparable with more demanding multi configurational methods, and in some cases much better than single determinantal schemes. Here we present the paradigmatic case of benzene.

Charge Transfer in the Electron Donor—Acceptor Complex BH3NH3

AbstractFor Abstract see ChemInform Abstract in Full Text.

Resonating valence bond wave function for the two-dimensional fractional spin liquid.

The unconventional low-lying spin excitations, recently observed in neutron scattering experiments on Cs2CuCl4, are explained with a spin liquid wave function. The dispersion relation as well as the wave vector of the incommensurate spin correlations are well reproduced within a projected BCS wave function with gapless and fractionalized spin-1/2 excitations around the nodes of the BCS gap function. The proposed wave function is shown to be very accurate for one-dimensional spin-1/2 systems and remains similarly accurate in the two-dimensional model corresponding to Cs2CuCl4, thus representing a good ansatz for describing spin fractionalization in two dimensions.

Charge transfer in the electron donor-acceptor complex BH3NH3.

As a simple yet strongly binding electron donor-acceptor (EDA) complex, BH(3)NH(3) serves as a good example to study the electron pair donor-acceptor complexes. We employed both the ab initio valence bond (VB) and block-localized wave function (BLW) methods to explore the electron transfer from NH(3) to BH(3). Conventionally, EDA complexes have been described by two diabatic states: one neutral state and one ionic charge-transferred state. Ab initio VB self-consistent field (VBSCF) computations generate the energy profiles of the two diabatic states together with the adiabatic (ground) state. Our calculations evidently demonstrated that the electron transfer between NH(3) and BH(3) falls in the abnormal regime where the reorganization energy is less than the exoergicity of the reaction. The nature of the NH(3)-BH(3) interaction is probed by an energy decomposition scheme based on the BLW method. We found that the variation of the charge-transfer energy with the donor-acceptor distance is insensitive to the computation levels and basis sets, but the estimation of the amount of electron transferred heavily depends on the population analysis procedures. The recent resurgence of interest in the nature of the rotation barrier in ethane prompted us to analyze the conformational change of BH(3)NH(3), which is an isoelectronic system with ethane. We found that the preference of the staggered structure over the eclipsed structure of BH(3)NH(3) is dominated by the Pauli exchange repulsion.

Ab initio VB Studies of the Ground and Low‐lying Excited States of BeH and BH

A scheme has been proposed to classify valence bond(VB) wave functions for the calculations of ground and excited states,according to the symmetry properties of one-electron orbitals which are involved in the construction of VB wave functions.This scheme is illustrated by the examples of BeH and BH.Ab initio VB computations of these two test molecules in combination with the present classification scheme give reliable results.For example,calculation results show that the state C2∑+of BeH is stable,with the bonding energy 0.87 eV and bond length 0.238nm,which are in good agreement with those obtained by Gerratt et al.The bonding features of ground and low-lying excited states of BeH and BH are discussed.

Effects of J-gate potential and uniform electric field on a coupled donor pair in Si for quantum computing

We present theoretical studies of a coupled-donor pair in Si via an unrestricted Hartree-Fock method with a generalized valence bond wave function. Polarization properties and exchange coupling for a phosphorous donor pair in silicon under a J-gate potential (modeled by a parabolic well) and a uniform electric field (either parallel or perpendicular to the interdonor axis) are examined. The energies and charge distributions of the lowest-lying singlet and triplet states as functions of the J-gate potential and uniform electric field for various donor separations are analyzed. Implications for Si:P electron-spin-based quantum computer architecture are discussed.

Electronic states and superconductivity in multilayer high-Tccuprates

We study electronic states of multilayer cuprates in the normal phases as functions of the number of ${\mathrm{CuO}}_{2}$ planes and the doping rate. The resonating valence bond wave function and the Gutzwiller approximation are used for a two-dimensional multilayer $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}J$ model. We calculate the electron-removal spectral functions at $(\ensuremath{\pi},0)$ in the ${\mathrm{CuO}}_{2}$ plane next to the surface to understand the angle-resolved photoemission spectroscopy (ARPES) spectra. We find that the trilayer spectrum is narrower than the bilayer spectrum but is wider than the monolayer spectrum. In the tri- and tetralayer systems, the outer ${\mathrm{CuO}}_{2}$ plane has different superconducting amplitude from the inner ${\mathrm{CuO}}_{2}$ plane, while each layer in the bilayer systems has same amplitude. The recent ARPES and NMR experiments are discussed in the light of the present theory.

Quasiparticle excitations in multi-layer cuprate superconductors

Quasiparticle excitations of multi-layer cuprates are studied in the normal phases. The electron-removal spectral functions at (π,0) in the CuO 2 planes nearest and second neighbor to the surface are calculated to understand the angle-resolved photoemission spectroscopy (ARPES) spectra. The resonating valence bond wave function and the Gutzwiller approximation are used for a two-dimensional multi-layer t– t ′– t″– J model. We find that the tri-layer spectrum is narrower than the bi-layer spectrum but is wider than the mono-layer spectrum. The recent ARPES experiments are discussed from the theoretical viewpoint.

An MO−VB Approach for the Determination of Intermolecular Forces. Theory and Calculations on the He2, He−CH4, and He−H2O Systems

We present the improvement of a previously developed strategy for the evaluation of intermolecular forces. The approach defines a variational VB (valence bond) wave function, consisting of single and double excitations from the SCF-MI (self-consistent field for molecular interactions) determinant. The central idea of the method is the determination of optimal virtual orbitals, to contract the virtual space spanned by all singly and doubly excited localized configurations, by means of an iterative optimization procedure. The performance of the strategy is tested by comparison with results where the full virtual space is considered, and the entire approach is also compared with more conventional quantum chemical methods. Test calculations on three weakly interacting complexes, namely, He2, He−CH4, and He−H2O, are presented. Whatever the system studied, we found an overall agreement between VB, MP4, and CCSD(T) results. The VB well depths estimates are somewhat larger than MP4 and CCSD(T) ones.

Introduction to Quantum Mechanics in Chemistry (Ratner, Mark A.; Schatz, George C.)

1. Introduction and Background to Quantum Mechanics. Aim of Theoretical Chemistry. Key Concepts from Classical Physics. Classical Mechanics. Classical Wave Theory. Early History of Quantum Mechanics. Particle Nature of Light. Wave Nature of Particles. Uncertainty Principle. Discovery of Quantum Mechanics. Concepts in Quantum Mechanics. 2. Quantum Theory. Postulates of Quantum Mechanics. Definitions of ...Y and ...|Y|2. Operators. Time-Dependent and Time-Independent Schrodinger Equations. Eigenvalues. Expectation Values. Properties of the Time-Independent Schrodinger Eigenfunctions. 3. Particle-in-Box Models. Particle in a One-Dimensional Box. Particle in a Two-Dimensional Box. Particle in a Three-Dimensional Box. Free-Electron Molecular Orbital Model. 4. Rigid-Rotor Models and Angular Momentum Eigenstates. Motions of a Diatomic Molecule: Separation of the Center of Mass. Rigid-Rotor Model in Two Dimensions. Three-Dimensional Rigid Rotor. Spherical Harmonics. Rotational Spectra. Angular Momentum. Dirac Notation. Raising and Lowering Angular-Momentum Operators. 5. Molecular Vibrations and Time-Independent Perturbation Theory. Diatomic Molecule Vibrations. Raising and Lowering Operators for the Harmonic Oscillator. Polyatomic Molecule Vibrations. Time-Independent Perturbation Theory. Examples. 6. The Hydrogen Atom. The Schrodinger Equation. Radial Solutions and Eigenvalues. Energy Eigenvalues Spectroscopy of the H Atom. Properties of Hydrogen and Hydrogenlike Wavefunctions. Atomic Units. 7. The Helium Atom. Schrodinger Equation. Independent-Particle Model. The Variational Method. Better Wavefunctions. 8. Electron Spin and the Pauli Principle. Electron Spin. The Pauli Principle. He-Atom Wavefunctions, Including Spin. Excited State of He. Energies of He(1s2s) States. Interaction of Electron Spin with Magnetic Fields. EPR and NMR. 9. Many-Electron Atoms. Many-Electron Hamiltonian and Schrodinger Equation. Slater Determinants. Hartree Method. Hartree-Fock Method. Koopmans' Theorem. Electron Correlation. Constants of the Motion. Angular-Momentum Operators for Many-Electron Atoms. Relativistic Effects. 10. Homonuclear Diatomic Molecules. Hydrogen Molecular Ion: Born-Oppenheimer Approximation. LCAO-MO Treatment of H2 +. Other H2 + States. Electronic Structure of Homonuclear Diatomics. Electronic Structure of H2: Molecular Orbital and Valence Bond Wavefunctions. Improvements to MO and VB Results for H2. 11. Ab Initio and Density Functional Methods. LCAO-MO-SCF Theory for Molecules. Atomic Orbitals. Hartree-Fock Calculations. Beyond Hartree-Fock. Density Functional Theory Methods. 12. Semiempirical Methods. Huckel Model. Extended Huckel Method. PPP Method. NDO Methods. 13. Applications of Group Theory. Group Theory for Point Groups. Applications of Group Theory to Molecular Quantum Mechanics. Symmetry Properties of Many-Electron Wavefunctions. Symmetry Properties of Molecular Vibrations. 14. Applications of Electronic Structure Theory. Potential-Energy Functions. Optimized Geometries and Frequencies. IR Spectra. Barriers to Reaction. Excited States. Molecular Clusters. Remarks on Other Methods. 15. Time Dependence and Spectroscopy. Transition Probabilities and the Golden Rule. Electronic Spectroscopy of Molecules. Vibration (Infrared) Spectroscopy. Appendices. Mathematical Background. Two-Electron Repulsion Integral. Character Tables. Atomic Units, Energy Conversion Factors, and Physical Constants. Solutions to Odd-Numbered Problems. Index.

Connections between coupled cluster and generalized valence bond theories

We explore the fundamental connections between certain approximate coupled cluster (CC) and generalized valence bond (GVB) wave functions. We show that the GVB restricted configuration interaction (GVB-RCI) wave function can be associated with a compact CC expansion in the valence space. However, careful analysis reveals that a standard CC expansion contains terms that are not contained in the GVB-RCI wave function. The offending terms violate an effective pairwise exclusion principle (PEP) that is present in the RCI expansion, but is not enforced in the CC analog. These terms do not affect the size separability of either method. Variational calculations show noticeable improvements to the CC wave function when the PEP is enforced, with the most significant improvements coming near the dissociation limit. We modify the standard CC amplitude equations by removing the PEP violating terms and demonstrate remarkably improved results for the dissociation of N2 and the double dissociation of H2O.

Ab initio non-orthogonal approaches to the computation of weak interactions and of localised molecular orbitals for QM/MM procedures

The research carried out by our group in the framework of the VB theory is reviewed. The modification of the Roothaan equations to avoid the basis set superposition error (BSSE) at the Hartree–Fock level of theory is presented. Significant applications of the resulting scheme, called Self Consistent Field for Molecular Interactions (SCF-MI), are summarised. The successive development of the SCF-MI method to properly take account of electron correlation effects by a compact multistructure size consistent Valence Bond (VB) wavefunction is described. The central idea of the method is that of expanding the orbitals in the partitioned basis sets located on each fragment so as to exclude BSSE in an a priori fashion, while taking properly into account orbital and geometry relaxation effects and the natural non orthogonality of the MO's of the interacting fragments. The very recent extension of the SCF-MI formulae to the determination of extremely localised molecular orbitals (ELMO) is also reviewed.

Resonating valence bond wave functions for strongly frustrated spin systems.

The resonating-valence-bond (RVB) theory for two-dimensional quantum antiferromagnets is shown to be the correct paradigm for large enough "quantum frustration." This scenario, proposed a long time ago but never confirmed by microscopic calculations, is strongly supported by a new type of variational wave function, which is extremely close to the exact ground state of the J(1)-J(2) Heisenberg model for 0.4 less than approximately J(2)/J(1) less than approximately 0.5. This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.