2.1 Principles of quantum mechanics

2.1.1 Wave-functions and operators

The theory of quantum mechanics is built upon the fundamental concepts of wave-functions and operators. The wave-function is a single-valued square-integrable function of the system parameters and time which provides a complete description of the system. Linear Hermitian operators act on the wave-function and correspond to the physical observables, those dynamical variables which can be measured, e.g. position, momentum and energy.

For systems of atomic nuclei^2.1 and electrons, which are the subject of this dissertation, the system parameters might be taken to be a set of position variables of the constituent particles (the notation adopted in this and the following chapters is to refer to electronic variables using a latin index and nuclear variables with a greek index) i.e. $\left\{ \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} \right\}$, their momenta $\left\{ \{ {\bf p}_i \} , \{ {\bf p}_\alpha \} \right\}$ or even a mixture of the two e.g. $\left\{ \{ {\bf r}_i \} , \{ {\bf p}_\alpha \} \right\}$. In contrast to a Newtonian system which is completely described by the positions and momenta of its constituents, the quantum-mechanical wave-function is a function of only one of these parameters per particle^2.2. The wave-function for the system is thus typically denoted by $\Psi \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} , t \right)$.

A notation due to Dirac [4] is often employed, which reflects the fact that this wave-function is simply one of many representations of a single state-vector in a Hilbert space, which is written as $\vert \Psi \rangle$, known as a ket. There also exists a dual space containing a set of bra vectors, denoted $\langle \Psi \vert$, defined by their scalar products and in one-to-one correspondence with the kets. The scalar product is written as a braket and is anti-linear in the first argument and linear in the second: thus $\langle \Psi \vert \Phi \rangle = (\langle \Phi \vert \Psi \rangle)^{\ast}$. It is worth noting here that state-vectors which differ only by a multiplicative non-zero complex constant describe the same state: we can thus restrict our interest to the set of normalised vectors defined such that the scalar product of the vector with its own conjugate equals unity:

$$\langle \Psi \vert \Psi \rangle = \int \prod_j {\mathrm d}{\... ...\left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} , t \right) = 1.$$ (2.1)

The operator corresponding to some observable $O$ is often written ${\hat O}$, and in general when this operator acts on some state-vector $\vert \Psi \rangle$, a different (not necessarily normalised) state-vector $\vert \Phi \rangle$ results:

$${\hat O} \vert \Psi \rangle = \vert \Phi \rangle.$$ (2.2)

However, for each operator there exists a set of normalised eigenstates, say $\left\{ \vert \chi_n \rangle \right\}$, which remain unchanged by the action of the operator i.e.

$${\hat O} \vert \chi_n \rangle = \lambda_n \vert \chi_n \rangle,$$ (2.3)

in which the constant $\lambda_n$ (always real for Hermitian operators) is the eigenvalue.

The postulates of quantum mechanics [5] state that for a system in state $\vert \Psi \rangle$:

• the outcome of a measurement of a dynamical variable is always one of the eigenvalues $\lambda_n$ of the corresponding operator,
• immediately following a measurement, the state-vector collapses to the eigenstate $\vert \chi_n \rangle$ corresponding to the measured eigenvalue^2.3,
• the probability of such a measurement^2.4 is
  

  $$P( \lambda_n ) = \left\vert \langle \chi_n \vert \Psi \rangle \right\vert^2 .$$ (2.4)

  
  

2.1.2 Expectation values

Much of the power of the theory comes from the fact that the quantum-mechanical states can be linearly superposed since this leads to no ambiguity in the action of linear operators^2.5. We now consider the quantity $\langle \Psi \vert {\hat O} \vert \Psi \rangle$. From Sturm-Liouville theory, the eigenstates of the operator ${\hat O}$ form a complete set, which means that any valid state-vector can be expressed as a linear superposition of those eigenstates with appropriate complex coefficients $\{c_n\}$:

$$\vert \Psi \rangle = \sum_n c_n \vert \chi_n \rangle .$$ (2.5)

These coefficients are easily obtained for Hermitian operators because the eigenstates are orthogonal (or can always be chosen to be orthogonal in the case of degenerate eigenvalues) which means that the scalar product of two different eigenstates vanishes:

$$\langle \chi_n \vert \chi_m \rangle = \delta_{nm} .$$ (2.6)

Either taking scalar products of both sides of equation 2.5 with the eigenstates $\{ \langle \chi_m \vert \}$, or by using the following concise expression of completeness;

$$\sum_n \vert \chi_n \rangle \langle \chi_n \vert = 1 ,$$ (2.7)

the expansion coefficients $\{c_n\}$ can be determined:

$$c_n = \langle \chi_n \vert \Psi \rangle ,$$ (2.8)

$$\vert \Psi \rangle = \sum_n \vert \chi_n \rangle \langle \chi_n \vert \Psi \rangle .$$ (2.9)

Now this result is applied to the quantity $\langle \Psi \vert {\hat O} \vert \Psi \rangle$:

$$\langle \Psi \vert {\hat O} \vert \Psi \rangle = \sum_m \left( \langle \chi_m \vert \Psi \rangle \right)^{\ast} \l... ..._m \vert {\hat O} \sum_n \vert \chi_n \rangle \langle \chi_n \vert \Psi \rangle$$

$$= \sum_n \lambda_n \left\vert \langle \chi_n \vert \Psi \rangle \right\vert^2$$ (2.10)

in which we have used the fact that ${\hat O}$ is linear, that the $\{ \vert \chi_n \rangle \}$ are eigenstates of ${\hat O}$ (2.3) and the orthonormality relation (2.6).

Since the only possible outcomes of a measurement of the observable $O$ corresponding to operator ${\hat O}$ are the eigenvalues $\{ \lambda_n \}$, with corresponding probabilities $\left\vert \langle \chi_n \vert \Psi \rangle \right\vert^2$ (2.4), the quantity $\langle \Psi \vert {\hat O} \vert \Psi \rangle$ is to be interpreted as the expectation value of $O$ for a system in state $\vert \Psi \rangle$. The normalisation condition $\langle \Psi \vert \Psi \rangle = 1$ corresponds to the condition that the probabilities sum to unity.

2.1.3 Stationary states

The final postulate of quantum mechanics states that between measurements, the state-vector evolves in time according to the time-dependent Schrödinger equation^2.6:

$${\hat H} \vert \Psi \rangle ={\mathrm i}\frac{\partial}{\partial t} \vert \Psi \rangle.$$ (2.11)

This treatment is non-relativistic: for heavy atoms there are significant relativistic effects but these can be incorporated a posteriori in the construction of the pseudopotentials (see 3.3). The operator ${\hat H}$ is known as the Hamiltonian and is the energy operator, which for systems of atomic nuclei and electrons takes the form

$${\hat H} = -\frac{1}{2} \sum_i \nabla_i^2 - \sum_{\alpha} \f... ...a}}{\left\vert {\bf r}_{\alpha} - {\bf r}_{\beta} \right\vert}$$ (2.12)

in which the nuclear masses $m_{\alpha}$ and atomic numbers $Z_{\alpha}$ appear. The first two terms on the right-hand side represent the kinetic energies of the electrons and nuclei respectively. The subsequent terms describe the electron-nuclear, electron-electron and inter-nuclear Coulomb interaction energies respectively.

Finally we note that if we solve the time-independent Schrödinger equation, the eigenvalue equation for the Hamiltonian, then the time-dependence of the wave-function takes a particularly simple form. The following separation of variables is made:

$$\Psi \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} , t \righ... ...eft( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} \right) \Theta (t)$$ (2.13)

which is successful and leads to the following equations, where $E$ is the separation constant:

$${\hat H} {\tilde \Psi}\left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} \right) = E {\tilde \Psi}\left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} \right) ,$$ (2.14)

$${\mathrm i} \frac{\mathrm d}{{\mathrm d} t} \Theta (t) = E \Theta (t) .$$ (2.15)

The ordinary differential equation 2.15 is straightforwardly solved, so that eigenfunctions of the Hamiltonian with energy $E$ take the form:

$$\Psi \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} , t \righ... ...bf r}_\alpha \} \right) \exp \left( -{\mathrm i} E t \right) .$$ (2.16)

States which are eigenfunctions of the Hamiltonian are also known as stationary states because the expectation values of time-independent operators for these states are also independent of time:

$$\langle \Psi \vert {\hat O} \vert \Psi \rangle = \int \prod_j {\mathrm d}{\bf r}_j~ \prod_{\beta} {\mathrm d}{\bf ... ...a \} \right) {\hat O} \Psi\left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} \right)$$

$$= \int \prod_j {\mathrm d}{\bf r}_j~\prod_{\beta}{\mathrm d}{\bf r}... ...\bf r}_i \} , \{ {\bf r}_\alpha \} \right) \exp \left( -{\mathrm i} E t \right)$$

$$= \langle {\tilde \Psi} \vert {\hat O} \vert {\tilde \Psi} \rangle .$$ (2.17)

From now on we shall be dealing with eigenstates of the Hamiltonian, and so will suppress the exponential time-dependence of the state and deal directly with the time-independent state $\vert {\tilde \Psi} \rangle$ instead.