Materials Science and Engineering/Doctoral review questions/Daily Discussion Topics/01122008

Variational Principle

Say you have a system for which you know what the energy depends on, or in other words, you know the Hamiltonian H. If one cannot solve the Schrödinger equation to figure out the ground state wavefunction, you may try any normalized wavefunction whatsoever, say φ, and the expectation value of the Hamiltonian for your trial wavefunction must be greater than or equal to the actual ground state energy. Or in other words:

$E_{ground}\leq \left\langle \phi |H|\phi \right\rangle$

This holds for any trial φ, and is obvious from the definition of the ground state wavefunction of a system. By definition, the ground state has the lowest energy, and therefore any trial wavefunction will have an energy greater than or equal to the ground state energy.

Proof

Your guessed wavefunction, φ, can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal):

$\phi =\sum _{n}c_{n}\psi _{n}\,$

Then, to find the expectation value of the hamiltonian:

$\left\langle \phi |H|\phi \right\rangle \,$	$=\left\langle \sum _{n}c_{n}\psi _{n}|H|\sum _{m}c_{m}\psi _{m}\right\rangle \,$
	$=\sum _{n}\sum _{m}\left\langle c_{n}\psi _{n}|E_{m}|c_{m}\psi _{m}\right\rangle \,$
	$=\sum _{n}\sum _{m}c_{n}^{*}c_{m}E_{m}\left\langle \psi _{n}|\psi _{m}\right\rangle \,$
	$=\sum _{n}|c_{n}|^{2}E_{n}\,$

Now, the ground state energy is the lowest energy possible, i.e. $E_{n}\geq E_{g}$. Therefore, if the guessed wave function φ is normalized:

$\left\langle \phi |H|\phi \right\rangle \geq E_{g}\sum _{n}|c_{n}|^{2}=E_{g}\,$

Hartree

The origin of the Hartree-Fock method dates back to the end of the 1920s, soon after the derivation of the Schrödinger equation in 1926. In 1927 D.R. Hartree introduced a procedure, which he called the self consistent field method, to calculate approximate wavefunctions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R.B. Lindsay and himself) set in the old quantum theory of Bohr. In the Bohr model of the atom, the energy of a state with principal quantum number n is given in atomic units as E = − 1 / n2. It was observed from atomic spectra that the energy levels of many-electron atoms were well-described by applying a modified version of Bohr's formula. By introducing the quantum defect d as an empirical parameter, the energy levels of a generic atom was well approximated by the formula E = − 1 / (n + d)2, in the sense that one could reproduce fairly well the observed transitions levels observed in the X-ray region. The existence of a non-zero quantum defect was attributed to electron-electron repulsion which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial screening of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data.

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. 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 Hartree's method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz (trial wavefunction) 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 wavefunction. Hartree's 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 ansatz 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 restricting 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 Fock Operator

Because the electron-electron repulsion term of the electronic molecular Hamiltonian involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation, (outlined under Hartree-Fock algorithm), all of the terms of the exact Hamiltonian except the nuclear-nuclear repulsion term are re-expressed as the sum of one-electron operators outlined below. The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature.

${\hat {F}}[\{\phi _{j}\}](1)={\hat {H}}^{core}(1)+\sum _{j=1}^{n/2}[2{\hat {J}}_{j}(1)-{\hat {K}}_{j}(1)]$

where:

${\hat {F}}[\{\phi _{j}\}](1)$

is the one-electron Fock operator generated by the orbitals $\phi _{j}$ ,

${\hat {H}}^{core}(1)=-{\frac {1}{2}}\nabla _{1}^{2}-\sum _{\alpha }{\frac {Z_{\alpha }}{r_{1\alpha }}}$

is the one-electron core Hamiltonian,

${\hat {J}}_{j}(1)$

is the Coulomb operator, defining the electron-electron repulsion energy due to the orbital of the j-th electron,

${\hat {K}}_{j}(1)$

is the exchange operator, defining the electron exchange energy. Finding the Hartree-Fock one-electron wavefunctions is now equivalent to solving the eigenfunction equation:

${\hat {F}}(1)\phi _{i}(1)=\epsilon _{i}\phi _{i}(1)$

where $\phi _{i}(1)$ are a set of one-electron wavefunctions, called the Hartree-Fock Molecular Orbitals.

Linear Combination of Atomic Orbitals

Typically, in modern Hartree-Fock calculations, the one-electron wavefunctions are approximated by a Linear combination of atomic orbitals. These atomic orbitals are called Slater-type orbitals. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more Gaussian-type orbitals, rather than Slater-type orbitals, in the interests of saving large amounts of computation time.

Various basis sets are used in practice, most of which are composed of Gaussian functions. In some applications, an orthogonalization method such as the Gram-Schmidt process is performed in order to produce a set of orthogonal basis functions. This can in principle save computational time when the computer is solving the Roothaan-Hall equations by converting the overlap matrix effectively to an identity matrix. However in most modern computer programs for molecular Hartree-Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the generalized eigenvalue problem, of which the Roothan-Hall equations are an example.

Hückel

The Hückel method or Hückel molecular orbital method (HMO) proposed by Erich Hückel in 1930, is a very simple LCAO MO Method for the determination of energies of molecular orbitals of pi electrons in conjugated hydrocarbon systems, such as ethene, benzene and butadiene. It is the theoretical basis for the Hückel's rule; the extended Hückel method developed by Roald Hoffmann is the basis of the Woodward-Hoffmann rules. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon, known in this context as heteroatoms.

Mathematics of the Hückel Method

The Hückel method can be derived from the Ritz method with a few further assumptions concerning the overlap matrix S and the Hamiltonian matrix H.

It is assumed that the overlap matrix S is the identity Matrix. This means that overlap between the orbitals is neglected and the orbitals are considered orthogonal. Then the generalised eigenvalue problem of the Ritz method turns into an eigenvalue problem.

The Hamiltonian matrix H = (Hij) is parametrised in the following way:

Hii = α for C atoms and α + hA β for other atoms A.

Hij = β if the two atoms are next to each other and both C, and kAB β for other neighbouring atoms A and B.

Hij = 0 in any other case

The orbitals are the eigenvectors and the energies are the eigenvalues of the Hamiltonian matrix. If the substance is a pure hydrocarbon the problem can be solved without any knowledge about the parameters. For heteroatom systems, such as pyridine, values of hA and kAB have to be specified.

Huckel Solution of Ethylene

In the Hückel treatment for ethylene, the molecular orbital $\Psi \,$ is a linear combination of the 2p atomic orbitals $\phi \,$ at carbon with their ratio's $c\,$:

$\ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}$

This equation is substituted in the Schrödinger equation:

$\ H\Psi =E\Psi$

with $H\,$ the Hamiltonian and $E\,$ the energy corresponding to the molecular orbital

to give:

$Hc_{1}\phi _{1}+Hc_{2}\phi _{2}=Ec_{1}\phi _{1}+Ec_{2}\phi _{2}\,$

This equation is multiplied by $\phi _{1}\,$ and integrated to give new set of equations:

$c_{1}(H_{11}-ES_{11})+c_{2}(H_{12}-ES_{12})=0\,$

$c_{1}(H_{21}-ES_{12})+c_{2}(H_{22}-ES_{22})=0\,$

where:

$H_{ij}=\int dv\phi _{i}H\phi _{j}\,$

$S_{ij}=\int dv\phi _{i}\phi _{j}\,$

All diagonal Hamiltonian integrals $H_{ii}\,$ are called coulomb integrals and those of type $H_{ij}\,$, where atoms i and j are connected, are called resonance integrals with these relationships:

$H_{11}=H_{22}=\alpha \,$

$H_{12}=H_{21}=\beta \,$

Other assumptions are that the overlap integral between the two atomic orbitals is 0

$S_{11}=S_{22}=1\,$

$S_{12}=0\,$

leading to these two homogeneous equations:

$c_{1}(\alpha -E)+c_{2}(\beta )=0\,$

$c_{1}\beta +c_{2}(\alpha -E)=0\,$

with a total of five variables. After converting this set to matrix notation:

${\begin{vmatrix}\alpha -E&\beta \\\beta &\alpha -E\\\end{vmatrix}}*{\begin{vmatrix}c_{1}\\c_{2}\\\end{vmatrix}}=0$

the trivial solution gives both wavefunction coefficients c equal to zero which is not useful so the other (non-trivial) solution is :

${\begin{vmatrix}\alpha -E&\beta \\\beta &\alpha -E\\\end{vmatrix}}=0$

which can be solved by expanding its determinant:

$\beta (\alpha -E)+(\alpha -E)^{2}-(\beta (\alpha -E)+\beta ^{2}=0\,$

$(\alpha -E)^{2}=\beta ^{2}\,$

$\alpha -E=\pm \beta \,$

or

$E=\alpha \pm \beta \,$

and

$\Psi =c_{1}(\phi _{1}\pm \phi _{2})\,$

After normalization the coefficients are obtained:

$c_{1}=c_{2}={\frac {1}{\sqrt {2}}},$

The constant β in the energy term is negative and therefore α + β is the lower energy corresponding to the HOMO and is α - β the LUMO energy.

Tight-Binding

In the tight binding model, it is assumed that the full Hamiltonian $H$ of the system may be approximated by the Hamiltonian of an isolated atom centred at each lattice point. The atomic orbitals $\psi _{n}$, which are eigenfunctions of the single atom Hamiltonian $H_{at}$, are assumed to be very small at distances exceeding the lattice constant. This is what is meant by tight-binding. It is further assumed that any corrections to the atomic potential $\Delta U$, which are required to obtain the full Hamiltonian $H$ of the system, are appreciable only when the atomic orbitals are small. A solution to the time-independent single electron Schrödinger equation $\Phi$ is then assumed to be a linear combination of atomic orbitals $\ {\psi }_{n}$

$\Phi ({\vec {r}})=\sum _{n,{\vec {R}}}b_{n,{\vec {R}}}\ \psi _{n}({\vec {r}}-{\vec {R}})$,

where n refers to the n-th atomic energy level and $^{\vec {R}}$ is an atomic site in the crystal lattice.

Using this approximate form for the wavefunction, and assuming only the m-th atomic energy level is important for the m-th energy band, the Bloch energies $\varepsilon _{m}$ are of the form

$\varepsilon _{m}({\vec {k}})=E_{m}-{\beta _{m}+\sum _{{\vec {R}}\neq 0}\gamma _{m}({\vec {R}})e^{i{\vec {k}}\cdot {\vec {R}}} \over {b_{m,{\vec {R}}}}+\sum _{{\vec {R}}\neq 0}\alpha _{m}({\vec {R}})e^{i{\vec {k}}\cdot {\vec {R}}}}$,

where $E_{m}$ is the energy of the $m$th atomic level,

$\beta _{m}=-\int \psi _{m}^{*}({\vec {r}})\Delta U({\vec {r}})\Phi ({\vec {r}})d{\vec {r}}$,

$\alpha _{m}({\vec {R}})=\int \psi _{m}^{*}({\vec {r}})\Phi ({\vec {r}}-{\vec {R}})d{\vec {r}}$,

and

$\gamma _{m}({\vec {R}})=-\int \psi _{m}^{*}({\vec {r}})\Delta U({\vec {r}})\Phi ({\vec {r}}-{\vec {R}})d{\vec {r}}$

are the overlap integrals.

The tight-binding model is typically used for calculations of electronic band structure and energy gaps in the static regime. However, in combination with other methods such as the random phase approximation (RPA) model, the dynamic response of systems may also be studied.

Bloch Theorem

A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle (usually, an electron) placed in a periodic potential. It consists of the product of a plane wave envelope function and a periodic function (periodic Bloch function) $u_{nk}(r)$ which has the same periodicity as the potential:

$\psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} ).$

The result that the eigenfunctions can be written in this form for a periodic system is called Bloch's theorem. The corresponding energy eigenvalue is Єn(k)= Єn(k + K), periodic with periodicity K of a reciprocal lattice vector. Because the energies associated with the index n vary continuously with wavevector k we speak of an energy band with band index n. Because the eigenvalues for given n are periodic in k, all distinct values of Єn(k) occur for k-values within the first Brillouin zone of the reciprocal lattice.

More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the different forms of the dynamical theory of diffraction.

The plane wave wavevector (or Bloch wavevector) k (multiplied by the reduced Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone. For a given wavevector and potential, there are a number of solutions, indexed by n, to Schrödinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete) band gap. The band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the independent electron approximation.

A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from things like imperfections that break the periodicity.

It can be shown that the eigenfunctions of a particle in a periodic potential can always be chosen this form by proving that translation operators (by lattice vectors) commute with the Hamiltonian. More generally, the consequences of symmetry on the eigenfunctions are described by representation theory.

The concept of the Bloch state was developed by Felix Bloch in 1928, to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by George William Hill (1877), Gaston Floquet (1883), and Alexander Lyapunov (1892). As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory (or occasionally the Lyapunov-Floquet theorem). Various one-dimensional periodic potential equations have special names, for example, Hill's equation:

${\frac {d^{2}y}{dx^{2}}}+\left(\theta _{0}+2\sum _{n=1}^{\infty }\theta _{n}cos(2nx)\right)y=0$

where the θ's are constants. Hill's equation is very general, as the θ-related terms may viewed as a Fourier series expansion of a periodic potential. Other much studied periodic one-dimensional equations are the Kronig-Penney model and Mathieu's equation.

Chemically Conjugated Systems

A chemically conjugated system is a system of atoms covalently bonded with alternating single and multiple (e.g. double) bonds (e.g., C=C-C=C-C) in a molecule of an organic compound. This system results in a general delocalization of the electrons across all of the adjacent parallel aligned p-orbitals of the atoms, which increases stability and thereby lowers the overall energy of the molecule.

The electron delocalization creates a region where electrons do not belong to a single bond or atom, but rather a group. An example would be phenol (C6H5OH, benzene with hydroxyl group) (diagramatically has alternating single and double bonds), which has a system of 6 electrons above and below the flat planar ring, as well as around the hydroxyl group.