Since, for any but the smallest of systems, a full VB calculation is out of the question, it is essential to devise a useful and systematic procedure for the arrangement of the bases and for the selection of a manageable subset of structures based upon these orbitals. These two problems are interrelated and cannot be discussed in complete isolation from one another, but we will consider the basis question first. In our two-electron calculations we have already addressed some of the issues, but here we look at the problems more systematically.

The AO bases

The calculations described in this section of the book have, for the most part, been carried out using three of the basis sets developed by the Pople school.

STO3G A minimal basis. This contains exactly the number of orbitals that might be occupied in each atomic shell.

6-31G A valence double-ζ basis. This basis has been constructed for atoms up through Ar.

6-31G* A valence double-ζ basis with polarization functions added. Polarization functions are functions of one larger l-value than normally occurs in an atomic shell in the ground state.

Any departures from these will be spelled out at the place they are used.

Our general procedure is to represent the atoms in a molecule using the Hartree–Fock orbitals of the individual atoms occurring in the molecule. (We will also consider the interaction of molecular fragments where the Hartree–Fock orbitals of the fragments are used.)

History

In physics and chemistry making a direct calculation to determine the structure or properties of a system is frequently very difficult. Rather, one assumes at the outset an ideal or asymptotic form and then applies adjustments and corrections to make the calculation adhere to what is believed to be a more realistic picture of nature. The practice is no different in molecular structure calculation, but there has developed, in this field, two different “ideals” and two different approaches that proceed from them.

The approach used first, historically, and the one this book is about, is called the valence bond (VB) method today. Heitler and London[8], in their treatment of the H2 molecule, used a trial wave function that was appropriate for two H atoms at long distances and proceeded to use it for all distances. The ideal here is called the “separated atom limit”. The results were qualitatively correct, but did not give a particularly accurate value for the dissociation energy of the H–H bond. After the initial work, others made adjustments and corrections that improved the accuracy. This is discussed fully in Chapter 2. A crucial characteristic of the VB method is that the orbitals of different atoms must be considered as nonorthogonal.

The other approach, proposed slightly later by Hund[9] and further developed by Mulliken[10] is usually called the molecular orbital (MO) method. Basically, it views a molecule, particularly a diatomic molecule, in terms of its “united atom limit”.

For many years chemists have considered that an understanding of the theory of the bonding of the homonuclear diatomic molecules from the second row of the periodic table is central to understanding all of bonding, and we consider these stable molecules first from our VB point of view. The stable molecules with interesting multiple bonds are B2, C2, N2, O2, and F2. Of course, F2 has only a single bond by ordinary bonding rules, but we include it in our discussion. Li2 is stable, but, qualitatively, is similar to H2. The question of the existence of Be2 is also interesting, but is really a different sort of problem from that of the other molecules. Of the five molecules we do consider, B2 and C2 are known only spectroscopically, while the other three exist at room temperature all around us or in the laboratory.

Atomic properties

Before we launch into the discussion of the molecules, we examine the nature of the atoms we are dealing with. As we should expect, this has a profound effect on the structure of the molecules we obtain. We show in Fig. 11.1 the first few energy levels of B through F with the ground state taken at zero energy. The L-S term symbols are also shown. The ground configurations of B and F each support just one term, P, but the other three support three terms. All of these are at energies below ≈4.2 eV (relative to their ground state energies).

One senses that it is out of style these days to write a book in the sciences all on one's own. Most works coming out today are edited compilations of others' articles collected into chapter-like organization. Perhaps one reason for this is the sheer size of the scientific literature, and the resulting feelings of incompetence engendered, although less honorable reasons are conceivable. Nevertheless, I have attempted this task and submit this book on various aspects of what is called ab initio valence bond theory. In it I hope to have made a presentation that is useful for bringing the beginner along as well as presenting material of interest to one who is already a specialist. I have taught quantum mechanics to many students in my career and have come to the conclusion that the beginner frequently confuses the intricacies of mathematical arguments with subtlety. In this book I have not attempted to shy away from intricate presentations, but have worked at removing, insofar as possible, the more subtle ones. One of the ways of doing this is to give good descriptions of simple problems that can show the motivations we have for proceeding as we do with more demanding problems.

This is a book on one sort of model or trial wave function that can be used for molecular calculations of chemical or physical interest. It is in no way a book on the foundations of quantum mechanics – there are many that can be recommended. For the beginner one can still do little better than the books by Pauling and Wilson[1] and Eyring, Walter, and Kimbal[2].

A variety of domain structures have been observed in metals undergoing (i) phase separation, or (ii) structural phase transitions [1]–[6]. In phase separation, a difference arises in the lattice constants of the two phases (lattice misfit). At a structural phase transition, anisotropically deformed domains of a stable low-temperature phase emerge in a quenched, metastable or unstable high-temperature phase. As a consequence, elastic strains are induced which radically influence the phase transition behavior. Here we will present Ginzburg–Landau theories for phenomena (i) and (ii) under the coherent condition [1], in which the lattice planes are continuous through the interface without any coherency loss due to dislocations, as illustrated in Fig. 10.1(a). In the incoherent case, however, dislocations are accumulated at the interface regions, and the resultant elastic effects have not yet been well investigated.

Among a number of important topics, here we cite examples of research on phase separation in binary alloys. In particular, experiments on Ni-base alloys are noteworthy [7]–[16]. (i) Figure 10.2 shows Ni3Al (γ′) cuboidal domains (precipitates) with the ordered L12 structure (illustrated in Fig. 3.10) in a disordered fcc Ni–Al alloy matrix [8]. Here, initially spherical domains changed their shapes into cuboids with facets in {100} planes as they grew. (ii) As can be seen in Fig. 10.3, at a very late stage cuboids can be seen sometimes to split into two plates or eight cuboids, despite an increase in the surface energy [3], [5a], [9, 15].

When an external parameter such as the temperature or the pressure is changed, physical systems in a homogeneous state often become unstable and tend to an ordered phase with broken symmetry [1]–[6]. The growth of order takes place with coarsening of domain or defect structures. Such ordering processes are observed in many systems such as spin systems, solids, and fluids. Historically, structural ordering and phase separation in alloys has been one of the central problems in metallurgy. These highly nonlinear and far-from-equilibrium processes have recently been challenging subjects in condensed matter physics. We will review various theories of phase ordering, putting emphasis on the dynamics of interfaces and vortices. As newly-explored examples we will discuss spinodal decomposition in one-component fluids near the gas–liquid critical point induced by the piston effect, that in binary fluid mixtures near the consolute critical point adiabatically induced by a pressure change, that under periodic quenching, and that in polymers and gels influenced by stress–diffusion coupling. A self-organized superfluid state will also be investigated, which is characterized by high-density vortices arising from competition between heat flow and gravity.

Phase ordering in nonconserved systems

Model A

We analyze the phase ordering in model A with a one-component order parameter (n = 1) given by (5.3.3)–(5.3.5).

To study equilibrium statistical physics, we will start with Ising spin systems (here-after referred to as Ising systems), because they serve as important reference systems in understanding various phase transitions [1]–[7]. We will then proceed to one- and two-component fluids with short-range interaction, which are believed to be isomorphic to Ising systems with respect to static critical behavior. We will treat equilibrium averages of physical quantities such as the spin, number, and energy density and then show that thermodynamic derivatives can be expressed in terms of fluctuation variances of some density variables. Simple examples are the magnetic susceptibility in Ising systems and the isothermal compressibility in one-component fluids expressed in terms of the correlation function of the spin and density, respectively. More complex examples are the constant-volume specific heat and the adiabatic compressibility in one- and two-component fluids. For our purposes, as far as the thermodynamics is concerned, we need equal-time correlations only in the long-wavelength limit. These relations have not been adequately discussed in textbooks, and must be developed here to help us to correctly interpret various experiments of thermodynamic derivatives. They will also be used in dynamic theories in this book. We briefly summarize equilibrium thermodynamics in the light of these equilibrium relations for Ising spin systems in Section 1.1, for one-component fluids in Section 1.2, and for binary fluid mixtures in Section 1.3.

Spin models

Ising hamiltonian

Let each lattice point of a crystal lattice have two microscopic states.

We will first give a theory of viscoelastic dynamics in polymeric binary systems, where a new concept of dynamic stress–diffusion coupling will be introduced in the scheme of viscoelastic two-fluid hydrodynamics. A Ginzburg–Landau theory of entangled polymer solutions will also be presented, in which chain deformations are represented by a conformation tensor. The reptation theory for entangled polymers will be summarized in Appendix 7A. We will also present a Ginzburg–Landau theory of gels to discuss dynamics and heterogeneities inherent to gels.

Viscoelastic binary mixtures

Entanglements among polymer chains impose severe topological constraints on the molecular motions. Their effects on polymer dynamics are now well described by the reptation theory in a surprisingly simple manner [1, 2]. In such systems, the stress relaxation takes place on a very long timescale τ (which should not be confused with the reduced temperature in near-critical systems). This means that a large network stress arises even for small deformations. If the timescale of the deformations is shorter than τ, the system behaves as a soft elastic body or gel. If it is longer than τ, we have a very viscous fluid.

In polymeric mixtures, it is highly nontrivial how the network stress acts on the two components and how it influences spatial inhomogeneities of the composition in various situations [3]–[5]. In this section we will introduce a mechanism of dynamical stress–diffusion coupling, which has recently begun to be recognized.

In this chapter we will introduce the simplest theory of phase transitions, the Landau theory [1]–[4]. It assumes a free energy H(ψ), called the Landau free energy, which depends on the order parameter ψ as well as the temperature and the magnetic field. The thermodynamic free energy F is the minimum of H(ψ) as a function of ψ. This minimization procedure gives rise to the mean field critical behavior. Historically, a number of mean field theories have been presented to explain phase transitions in various systems. They reduce to the Landau theory near the critical point. Examples we will treat are the Bragg–Williams theory [5] for Ising spin systems and alloys undergoing order–disorder phase transitions, the van der Waals theory of the gas-liquid transition [6], the Flory–Huggins theory and the classical rubber theory for polymers and gels. We will also discuss tricritical phenomena in the scheme of the Landau theory. In Appendix 3A elastic theory for finite strain will be considered, which will be needed to understand the volume-phase transition in gels.

Landau theory

Order parameter and constrained free energy

It is desirable to sum up the spin configurations in (1.1.9) to exactly determine the thermodynamic limit. This attempt has not been successful for the 3D Ising model, while it was successful for 2D and is a simple exercise for 1D [3].

Slow collective motions in physical systems, particularly those near the critical point, can be best described in the framework of Langevin equations. We may set up Langevin equations when the timescales of slow and fast dynamical variables are distinctly separated. This framework originates from the classical Brownian motion and is justified microscopically via the projection operator formalism. First, in Sections 5.1–5.2, these general aspects will be discussed with a summary of the projection operator method in Appendix 5B. Second, in Section 5.3, we will examine simple Langevin equations in critical dynamics (models A, B, and C) and introduce dynamic renormalization group theory. These models have been used extensively to study fundamental problems in critical dynamics and phase ordering. Third, in Section 5.4, we will review the general linear response theory, putting emphasis on linear response to thermal disturbances.

Langevin equation for a single particle

Brownian motion

Most readers will be aware of the zig-zag motions of a relatively large particle, calleda Brownian particle, suspended in a fluid. When its mass m0 is much larger than those of the surrounding particles, appreciable changes of the velocity of the Brownian particle can be caused as a result of a large number of collisions with the surrounding molecules.

In this chapter we will present the Ginzburg–Landau–Wilson (GLW) hamiltonian and briefly explain the renormalization group (RG) theory in the scheme of the ∈ = 4 – d expansion [1]–[12]. As unique features in this book we will introduce a subsidiary energy-like variable in addition to the order parameter, discuss GLW models appropriate for fluids, and derive a simple expression for the thermodynamic free energy consistent with the scaling theory and the two-scale-factor universality. We will try to reach the main RG results related to observable quantities in the simplest and shortest way without too much formal argument. In practice, such an approach is needed for those whose main concerns are advanced theories of dynamics. Furthermore, we will discuss inhomogeneous two-phase coexistence and the surface tension near the critical point, near the symmetrical tricritical point, and in polymer solutions and blends. In addition, we will examine vortices in systems with a complex order parameter. These topological defects are key entities in phase-ordering dynamics discussed in Chapters 8 and 9.

Ginzburg–Landau–Wilson free energy

Gradient free energy

When the order parameter ψ changes slowly in space, the simplest generalization of the Landau free energy is of the form, which is called the Ginzburg–Landau–Wilson (GLW) hamiltonian.