Applications of quantum mechanics

Applications of quantum mechanics: Chemical bonds, molecules and complex substances

Linus Pauling began his work on the chemical bond, figuring calculations and comparing his results to existing experimental data. He affirmed that Heitler and London’s work meshed comfortably with G. N. Lewis‘ theory of the shared electron pair and he published articles on the subject, in the process introducing many chemists to the notion of using quantum mechanics as a tool for the study of non-physics problems. In early 1928, he suggested that quantum mechanics could answer the question of carbon bonding – a revolutionary idea at the time. Unfortunately, while the preliminary mathematics were promising, the sheer mathematical computing power required did not exist for Pauling to fully solve the problem.

In 1930 M.I.T. physicist John C. Slater succeeded in simplifying Schrödinger’s mathematical description of the types of changes experienced by any quantum system over time — an important mathematical model known as the Schrödinger Wave Equation. By slightly restructuring Slater’s set of simplified equations, Pauling was able to utilize the concept of wave functions to describe new orbitals that matched the known traits of the carbon-tetrahedron bond. Not only did these new methods allow Pauling to calculate the data for basic tetrahedral bonds, they also provided stable footing for detailing the precise structures of a series of complex molecules. This was the genesis of valence-bond theory — a hugely important marriage of quantum physics and structural chemistry. Complex, in chemistry, a substance, either an ion or an electrically neutral molecule, formed by the union of simpler substances (as compounds or ions) and held together by forces that are chemical (i.e., dependent on specific properties of particular atomic structures) rather than physical. The formation of complexes has a strong effect on the behaviour of solutions. See also chemical association; coordination compound.

Part of Pauling's work on the nature of the chemical bond led to his introduction of the concept of orbital hybridization. While it is normal to think of the electrons in an atom as being described by orbitals of types such as s and p, it turns out that in describing the bonding in molecules, it is better to construct functions that partake of some of the properties of each. Thus the one 2s and three 2p orbitals in a carbon atom can be (mathematically) 'mixed' or combined to make four equivalent orbitals (called sp3 hybrid orbitals), which would be the appropriate orbitals to describe carbon compounds such as methane, or the 2s orbital may be combined with two of the 2p orbitals to make three equivalent orbitals (called sp2 hybrid orbitals), with the remaining 2p orbital unhybridized, which would be the appropriate orbitals to describe certain unsaturatedcarbon compounds such as ethylene. Other hybridization schemes are also found in other types of molecules. Another area which he explored was the relationship between ionic bonding, where electrons are transferred between atoms, and covalent bonding, where electrons are shared between atoms on an equal basis. Pauling showed that these were merely extremes, and that for most actual cases of bonding, the quantum-mechanical wave function for a polar molecule AB is a combination of wave functions for covalent and ionic molecules. Here Pauling's electronegativity concept is particularly useful; the electronegativity difference between a pair of atoms will be the surest predictor of the degree of ionicity of the bond.

In early 1931, Pauling released a paper detailing six rules, later known as “Pauling’s Rules,” that dictated the basic principles governing the molecular structure of any given molecule. He presented his findings in the simplest language possible, avoiding complex mathematics in order to make the concepts accessible to his fellow chemists. This paper, of course, was titled “The Nature of the Chemical Bond” and would serve as the basis for his immensely popular textbook of the same name.

The third of the topics that Pauling attacked under the overall heading of "the nature of the chemical bond" was the accounting of the structure of aromatic hydrocarbons, particularly the prototype, benzene. The best description of benzene had been made by the German chemist Friedrich Kekulé. He had treated it as a rapid interconversion between two structures, each with alternating single and double bonds, but with the double bonds of one structure in the locations where the single bonds were in the other. Pauling showed that a proper description based on quantum mechanics was an intermediate structure which was a blend of each. The structure was a superposition of structures rather than a rapid interconversion between them. The name "resonance" was later applied to this phenomenon.In a sense, this phenomenon resembles those of hybridization and also polar bonding, both described above, because all three phenomena involve combining more than one electronic structure to achieve an intermediate result.

Edward Teller was a Hungarian-American theoretical physicist who is known colloquially as "the father of the hydrogen bomb", although he claimed he did not care for the title.[1] He made numerous contributions to nuclear and molecular physics, spectroscopy (in particular the Jahn–Teller and Renner–Teller effects), and surface physics. in 1951 Teller and Ulam made a breakthrough, and invented a new design, proposed in a classified March 1951 paper, On Heterocatalytic Detonations I: Hydrodynamic Lenses and Radiation Mirrors, for a practical megaton-range H-bomb. The exact contribution provided respectively from Ulam and Teller to what became known as the Teller–Ulam design is not definitively known in the public domain, and the exact contributions of each and how the final idea was arrived upon has been a point of dispute in both public and classified discussions since the early 1950s.

Paul Teller also presents the basic ideas of quantum field theory in a way that is understandable to readers who are familiar with non-relativistic quantum mechanics. He provides information about the physics of the theory without calculational detail, and he enlightens readers on how to think about the theory physically. Along the way, he dismantles some popular myths and clarifies the novel ways in which quantum field theory is both a theory about fields and about particles. His goal is to raise questions about the philosophical implications of the theory and to offer some tentative interpretive views of his own. Quantum field theory (QFT) started in the late 1920s soon after quantum mechanics (QM) was developed.’ It was motivated by the desire to generalize QM so as to become consistent with the special theory of relativity, as well as by the fact that QM became increasingly unable to account for experimental results of improved accuracy (effects of spin and fast motion). In QM the motion of particles is restricted to speeds that are small compared to that of light; it is the quantum form of Newtonian mechanics. But electromagnetic theory is an intrinsically relativistic theory. The quantum mechanical treatment of particles in electromagnetic interaction (the basis of atomic physics) is therefore strictly inconsistent from a symmetry point of view. (It does not have a consistent symmetry under space-time transformations). A new relativistic QM was therefore to be developed that would be valid for all speeds v ≪ c. And, within that new theory, quantum electrodynamics (the interaction of relativistic electrons with electromagnetic fields) would be of special interest being a more accurate account of atomic physics.

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be separated. The approach is named after Max Born, and J. Robert Oppenheimer. In mathematical terms, it allows the wavefunction of a molecule to be broken into its electronic and nuclear (vibrational, rotational) components.

In the first step of the BO approximation the electronic Schrödinger equation is solved, yielding the wavefunction ψ_(electronic) depending on electrons only. For benzene this wavefunction depends on 126 electronic coordinates. During this solution the nuclei are fixed in a certain configuration, very often the equilibrium configuration. If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be in nuclear coordinates. In the second step of the BO approximation this function serves as a potential in a Schrödinger equation containing only the nuclei—for benzene an equation in 36 variables.

The success of the BO approximation is due to the difference between nuclear and electronic masses. The approximation is an important tool of quantum chemistry: all computations of molecular wavefunctions for large molecules make use of it, and without it only the lightest molecule, H2, can be handled. Even in the cases where the BO approximation breaks down, it is used as a point of departure for the computations.

The electronic energies consist of kinetic energies, interelectronic repulsions, internuclear repulsions, and electron–nuclear attractions. In accord with the Hellmann-Feynman theorem, the nuclear potential is taken to be an average over electron configurations of the sum of the electron–nuclear and internuclear electric potentials.

In molecular spectroscopy, because the ratios of the periods of the electronic, vibrational and rotational energies are each related to each other on scales in the order of a thousand, the Born–Oppenheimer name has also been attached to the approximation where the energy components are treated separately.

E_total = E_eletronic + E_vibrational + E_rotational + E_nuclear

The nuclear spin energy is so small that it is normally omitted.

The Born-Oppenheimer approximation is one of the basic concepts underlying the description of the quantum states of molecules. This approximation makes it possible to separate the motion of the nuclei and the motion of the electrons. This is not a new idea for us. We already made use of this approximation in the particle-in-a-box model when we explained the electronic absorption spectra of cyanine dyes without considering the motion of the nuclei. Then we discussed the translational, rotational and vibrational motion of the nuclei without including the motion of the electrons. In this chapter we will examine more closely the significance and consequences of this important approximation. Note, in this discussion nuclear refers to the atomic nuclei as parts of molecules not to the internal structure of the nucleus.

The Born-Oppenheimer approximation neglects the motion of the atomic nuclei when describing the electrons in a molecule. The physical basis for the Born-Oppenheimer approximation is the fact that the mass of an atomic nucleus in a molecule is much larger than the mass of an electron (more than 1000 times). Because of this difference, the nuclei move much more slowly than the electrons. In addition, due to their opposite charges, there is a mutual attractive force of

(Ze²)/r²

acting on an atomic nucleus and an electron for z is the atomic number, e is the electron charge and r is the dintance between the electron and the nuclei. This force causes both particles to be accelerated. Since the magnitude of the acceleration is inversely proportional to the mass, a = F/m, the acceleration of the electrons is large and the acceleration of the atomic nuclei is small; the difference is a factor of more than 2000. Consequently, the electrons are moving and responding to forces very quickly, and the nuclei are not.

The Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

The Hartree–Fock method often assumes that the exact, N-body wave function of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of N spin-orbitals. By invoking the variational method, one can derive a set of N-coupled equations for the N spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system.

Especially in the older literature, the Hartree–Fock method is also called the self-consistent field method (SCF). In deriving what is now called the Hartree equation as an approximate solution of the Schrödinger equation, Hartree required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear Hartree–Fock equations also behave as if each particle is subjected to the mean field created by all other particles (see the Fock operator below) and hence, the terminology continued. The equations are almost universally solved by means of an iterative method, although the fixed-point iteration algorithm does not always converge. This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method.

The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics. (See Hartree–Fock–Bogoliubov method for a discussion of its application in nuclear structure theory). In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state. For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately.

The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case. The discussion here is only for the Restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) doubly occupied. Open-shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree–Fock methods:

• Restricted open-shell Hartree–Fock (ROHF)
  
• Unrestricted Hartree–Fock (UHF)
  

Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e., ab initio. His first proposed method of solution became known as the Hartree method or Hartree product. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle functions.

In 1930, Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function. The Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of quantum statistics.

It was then shown that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable initial placement of a tool at a work piece for applying the variational principle. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. Fock's original method relied heavily on group theory and was too abstract for contemporary physicists to understand and implement. In 1935, Hartree reformulated the method more suitably for the purposes of calculation.

The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the central field approximation, to impose that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. Even so, solution by hand of the Hartree–Fock equations for a medium-sized atom were laborious; small molecules required computational resources far beyond what was available before 1950.

The Hartree–Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule as described in the Born–Oppenheimer approximation. Since there are no known solutions for many-electron systems (there are solutions for one-electron systems such as hydrogenic atoms and the diatomic hydrogen cation), the problem is solved numerically. Due to the nonlinearities introduced by the Hartree–Fock approximation, the equations are solved using a nonlinear method such as iteration, which gives rise to the name "self-consistent field method."

Approximations:

• The Hartree–Fock method makes five major simplifications in order to deal with this task:
  
• The Born–Oppenheimer approximation is inherently assumed. The full molecular wave function is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
  
• Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic.
  
• The variational solution is assumed to be a linear combination of a finite number of basis functions, which are usually (but not always) chosen to be orthogonal. The finite basis set is assumed to be approximately complete.
  
• Each energy eigenfunction is assumed to be describable by a single Slater determinant, an antisymmetrized product of one-electron wave functions (i.e., orbitals).
  
• The mean field approximation is implied. Effects arising from deviations from this assumption are neglected. These effects are often collectively used as a definition of the term electron correlation. However, the label "electron correlation" strictly spoken encompasses both Coulomb correlation and Fermi correlation, and the latter is an effect of electron exchange, which is fully accounted for in the Hartree–Fock method. Stated in this terminology, the method only neglects the Coulomb correlation. However, this is an important flaw, accounting for (among others) Hartree-Fock's inability to capture London dispersion.
  

Relaxation of the last two approximations give rise to many so-called post-Hartree–Fock methods.

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