The classical theory of chemical valence, first, is naturally formalized in mathematics in the area of graph theory and, second, finds an extension in quantum mechanics in terms of the Heitler–London–Pauling “valence-bond” (VB) theory. Thus, VB theory stands in a fairly unique position, although in quantum chemistry, there often has been a preference for the alternative (perhaps even “complementary”) molecular orbital (MO) theory, presumably in large part because of computational efficacy for general molecular structures. Indeed, as formulated by Pauling and others, VB theory was described as a configuration interaction (CI) problem when there were multiple relevant classical valence structures for the same molecular structure. Also, as now recognized, a direct assault on CI is computationally intensive, prone to size-inconsistency problems, and effectively limited to smaller systems—whereas indirect approaches, e.g., via wave-function cluster expansions or renormalization-group theory, often neatly avoid these problems. Thus, what is (and perhaps always has been) needed is “many-body” schemes for VB computations (as well as for higher-order MO-based approaches, too). Here, then, certain such many-body VB-amenable computational schemes are to be discussed, in the context of semiempirical (explicitly correlated) graphical models. The collection of models are described and interrelated in a fairly comprehensive systematic manner. A selection of many-body cluster expansion methods are then discussed with special reference to resonating VB wave functions and the fundamental graph-theoretic nature of the consequent problems (such as also are noted to arise in lattice-discretized statistical-mechanical problems, too). Some examples are described incorporating resonance among exponentially great numbers of VB structures as applied: for large icosahedral-symmetry fullerenic structures, for the (polyacetylenic) linear chain, and for ladderlike conjugated polymers. It is contended that practicable many-body VB-theoretic methods are now available, retaining clear links to classical chemical valence theory. Hopefully, too, these methods may soon find use beyond the semi-empirical framework. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 421–438, 1997
Abstract ..................................................................................................... 2 1.1 Introduction ...................................................................................... 3 1.2 Valence Bond Theory ....................................................................... 5 1.2.1 Adiabatic Approximation ..................................................... 5 1.2.2 The Virial Theorem on the Molecular Systems .................... 9 1.2.3 The Chemical Bond Through the Valence Bond Method (VBM) ................................................................... 12 1.3 Molecular Orbital Theory of Bonding ........................................... 22 1.4 Sub-Quantum Chemistry: Chemical Bonding Field and Bondons ......................................................................... 34 1.4.1 Introducing Chemical Field ................................................ 35 1.4.2 Predicting the Bondonic’s Existence .................................. 38 1.4.3 Chemical Bonding by Bondons ......................................... 45 1.4.4 Chemical Bonding as a Bosonic Quantum Condensate ..... 55 1.4.4.1 ψ-Theory of Bose-Einstein Condensation ........... 55 1.4.4.2 DFT adapted to Bose-Einstein Condensation ...... 58 1.4.4.3 Local Density Approximation of Bose-Einstein Condensation ................................ 61 1.4.4.4 Heitler-London Bondonic Condensation ............. 64 1.4.4.5 Molecular Orbitals’ Bondonic Condensation ...... 70 1.5 Conclusion ..................................................................................... 82 Keywords ................................................................................................ 84 References ............................................................................................... 85 Author’s Main References ............................................................. 85 Specific References ........................................................................ 85 ABSTRACT The classical quantum methods in characterizing chemical bonding, i.e., the valence orbital theory (of Pauling) and the molecular orbital theory (of Heitler and London) are firstly exposed (and illustrated at least for H2 system) for didactical and benchmarking research purpose on fundamental nature of the chemical bonding. Then the “missing link” in characterizing chemical bonding field by its associated quantum particle, the bondon B , is revealed by employing the combined Bohmian quantum formalism within the Schrödinger quantum picture of electron motions and characterized by its mass (mB), velocity (vB), charge (eB), and life-time (t B); the mass-velocity-charge-time quaternion properties of bondons’ particles were used in discussing various paradigmatic types of chemical bond towards assessing their covalent, multiple bonding, metallic and ionic features. The bondonic picture was completed by discussing the relativistic charge and life-time (the actual zitterbewegung) problem, i.e., showing that the bondon equals the benchmark electronic charge through moving with almost light velocity. It carries negligible, although non-zero, mass in special bonding conditions and towards observable femtosecond life-time as the bonding length increases in the nanosystems and bonding energy decreases according with the bonding length-energy relationship of Heisenberg type E kcal mol X Abond bond[ / ] [ ]× = 0 182019, providing this way the predictive framework in which the B particle may be observed when either associate Ebond or length Xbond is known. However, being the bondon a boson, as the particle carrying the quantum information of the chemical wave/orbital/ field, further, possibility in assessing the density functional theory (DFT) for of the Kohn-Sham equation within Bose-Einstein condensation (BEC) and of its further reduction to the nonlinear Schrodinger equation under the GrossPitaevsky equation is responded in positive through employing specific DFTBEC exchange functional combined within the Hartree-Fock-Bogoliubov mean-field theory. This non-linear generalization of the Schrödinger equation is then employed in first providing the DFT-BEC connection at the level of the Thomas-Fermi approximation; such connections are further used towards generalizing the classical Heitler-London formalism and bonding-antibonding equations of homopolar chemical bonding with the aid of the mass quantification of the quantum particle of the chemical bonding field – the bondons. Actually, two branches of bosonic-bondonic condensate were identified as physical and chemical bonding BEC, both with bonding and antibonding features. These findings innovatively complete the already “classical” quantum treatment of chemical bonding through the valence orbital theory (dealing with resonances of molecular structures) and then to the orbital molecular theory (providing the hybridization and the bonding/anti-bonding paradox), here also exposed for their main features.
Studying complex molecular systems, whether in the gas or a condensed phase, requires highly accurate quantum chemistry methods whose computational effort remains tractable when treating realistic systems. Commonly used ab initio methods such as Coupled Cluster (CC) or Density Functional Theory (DFT) generally provide reliable results when the underlying approximations and parameterizations are appropriate for the systems being studied. Among the less popular quantum chemistry methods, Valence Bond (VB) theory and Quantum Monte Carlo (QMC) methods, each provides different unique advantages and disadvantages. Ab initio VB, while usually only qualitative for energetic properties such as enthalpies of formation, has the distinct advantage of providing very compact wave functions that can be readily mapped into the intuitive Lewis structures that chemists very often use when qualitatively describing the valence electronic structure of atoms and molecules. QMC using trial wave functions based on multireference ab initio quantum methods can give some of the most accurate energies and related properties, however, this is often at a great computational cost. Furthermore, the large wave function expansions in the trial function are often difficult to qualitatively interpret. In this lecture, we will present the Valence-Bond Quantum Monte Carlo methods (VB-QMC) which retain the interpretability of VB and the high accuracy of QMC. We will show a few applications which demonstrate their accuracy for molecules in the gas phase and we will examine areas where innovations could be made.
Electronic Correlation in Molecules - Some Fundamental Problems Improving the Panoply of Quantum Chemical Methods for the Study of Correlated Extended Systems Lie-Algebraic Structure and the Coupled Cluster Method Configuration Interaction Studies Using Biorthogonal Approach to VB Basis Optimised Virtual Orbitals for the Calculation of Molecular Properties of Excited Electronic States A Novel MO-VB Scheme for Excitation Gaps in Correlated Mode Hamiltonians Correlated Approach to Molecular Properties Infinite Volume Extrapolations of Finite Clusters Calculations - How Correct are These? Calculation of the Hyperpolarisabilities of Organic Molecules and their Role in the Interpretation of Nonlinear Optical Phenomena in Organic Crystals Energy Thresholds in Polysilanes - Elementary Excitations of Alternating Conjugated Chains Inter- and Intramolecular Spin Coupling in Organics Electron Correlation Effects in Semiconductorquantum Dot Nanostructures Numerical Simulations of Strongly-Correlated Systems-Interlayer Tunnelling Mechanism, Spin-Charge Separation and Odd-Gap Pairing Metal Insulator Transition in 3rd Transition Metal Oxides Superconductivity in Doped Fullerenes.
Many-body perturbation theories (MBPT) (both single and multireference), by design, are capable of representing electron correlation very well. However, the practical implementation of these methods for many-electron systems beyond a certain order is extremely difficult, if not impossible. For instance, there are hardly any calculations beyond third order at the multireference and fifth order at the single reference of perturbation theories. Thus, these methods are typically limited to systems where the higher-order electron correlation effect is not significant. A practical solution to this problem is the coupled cluster theory, which was first introduced by Coester [20] and Coester and Kümmel [21] in nuclear physics and was later adopted in the electron structure theory by Čížek [23, 106] and Paldus and Čížek [107]. The coupled-cluster (CC) theory is a non-perturbative all-order theory. This stateof-the-art theory has now been recognized as the basis of modern quantitative quantum chemistry and atomic physics for high-precision calculations. Its superior accuracy is rooted in the exponential ansatz for the wave function, which provides an energy expression that scales correctly with the number of electrons and under certain provisions leads to a faster convergence of the many-body expansion.
In this book, we encounter the hierarchy problem when we apply the equation-of-motion technique to analyze the quantum dynamics of the coupled light–matter system. To obtain systematic approximations, we use the so-called cluster-expansion method where many-body quantities are systematically grouped into cluster classes based on how important they are to the overall quantum dynamics. With increasing complexity, the clusters contain independent single particles (singlets), interacting pairs (doublets), three-particle terms (triplets), coupled four-particle contributions (quadruplets), as well as higher-order correlations. In this context, the N -particle concept is somewhat formal because it refers to generic N -particle expectation values 〈 N 〉, which may consist of an arbitrary mixture of carrier, photon, and phonon operators, as discussed in Section 14.2. In order to truncate the hierarchy problem at a given level, 〈 N 〉 is approximated through a functional structure that includes all clusters up to the predetermined level while all remaining clusters with a higher rank are omitted. It is natural that the corresponding approximations can be systematically improved by increasing the number of clusters included. To the best of our knowledge, the idea of coupled-clusters approaches was first formulated by Fritz Coester and Hermann Kümmel in the 1950s to describe nuclear many-body phenomena. The approach was then modified for the needs of quantum chemistry by Jiri Cizek 1966 to deal with many-body phenomena in atoms and molecules. Currently, it is one of the most accurate methods to compute molecular eigenstates.
The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid
He3, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$.
F is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the
f(rij) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements
Hmn,
Nmn of the Hamiltonian and identity operators with respect to the correlated basis ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the
Hmn are classified according to order of magnitude in the quantity $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (
Hmn) and (
Nmn), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (
Hmn) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (
Hmn). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in $\ensuremath{\rho}\ensuremath{\omega}$. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (
Hmn). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, applied to the given singular Hamiltonian, leads---for any
F in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in $\ensuremath{\rho}\ensuremath{\omega}$, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},
Q-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid
He3 has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.