Mathematics for quantum chemistry

Introduction
Quantum mechanics grew up from two different points of view, which represent two analogous mathematical formulations of eigenvalue problems. One is the wave mechanics of Schrodinger. In wave mechanics, operators are differential expressions, such as $\dfrac{d}{dx}$, and the eigenvalue equation takes the form of a differential equation, and relies on the calculus for its solution. The other formulation is the matrix mechanics of Heisenberg, in which operators are represented by algebraic entities called matrices; instead of a function in the eigenvalue equation, the matrix operator operates on a vector $\zeta$ to transform $\zeta$ into a vector parallel to $\zeta$, but $q$ times as long: $$Q\zeta=q\zeta.$$ This is the matrix-mechanical formulation of the eigenvalue problem. The solution of this form of the eigenvalue problem relies on algebra.

These two approaches to quantum mechanical problems are deeply interrelated; the work of Dirac shows the underlying equivalence of the two points of view, as well as of the corresponding mathematical techniques. [...]

Orthogonal functions
Definition: The expansion interval is the range of the independent variables assumed by the functions under consideration.

Definition: The inner product of the two complex-valued functions $f$ and $g$ of a continuous variable on their expansion interval $[a,b]$ is $$\langle f|g \rangle=\int_a^b f(x)^*g(x)dx.$$
The order is not important for real-valued functions but important for complex-valued functions: $$\begin{align}\langle g|f \rangle&=\int g(x)^*f(x)dx\\&=\big(\int f(x)^*g(x)dx\big)^*\\ &=\langle f|g \rangle^*.\end{align}$$

Definition: Two functions $f(x)$ and $g(x)$ are said to be orthogonal on the interval $[a,b]$ if their inner product on $[a,b]$ is zero: $$\langle f|g \rangle=\int_a^b f^*g=0=\int_a^b g^*f=\langle g|f \rangle.$$

Definition: The norm of a function on the interval $[a,b]$ is the inner product of the function with itself, and may be symbolised by $N$: $$N(f)=\langle f|f \rangle=\int_a^b f^*g=0=\int_a^b f^*f.$$

The norm of a function is a real, positive quantity: $$f^*f=(\text{Re}\:f-i\text{Im}\:f)(\text{Re}\:f+i\text{Im}\:f)=(\text{Re}\:f)^2+(\text{Im}\:f)^2.$$

Definition: A function is said to be normalised if its norm is one: $\langle f|f \rangle=1$.
Suppose $f$ has a norm $N$. Then the function $\dfrac{f}{\sqrt{N}}$ will have a norm of one, since $$\langle \dfrac{f}{\sqrt{N}}|\dfrac{f}{\sqrt{N}} \rangle=\dfrac{1}{N}\langle f|f \rangle=\dfrac{N}{N}=1.$$
Reminder: The expansion interval must be specified before a statement about orthogonality or normality can be made. On $[-1,1]$, $\langle x|x^2 \rangle=\int_{-1}^1 x^*x^2 dx=\int_{-1}^1 x^3 dx=\dfrac{x^4}{4}|_{-1}^1=0$, so we can say that on $[-1,1]$, $x$ and $x^2$ are orthogonal functions. However, on $[0,1]$, $\langle x|x^2 \rangle=\int_0^1 x^3 dx=\dfrac{x^4}{4}|_0^1=\dfrac{1}{4}\neq 0$, so the functions are not orthogonal in this interval.

Definition: $\{F_i\}$ is a complete set of functions such that any other function $f$ may be expressed as a linear combination of members of the set $\{F_i\}$ on a prescribed expansion interval. [Namely, if the set $\{F_i\}$ is complete, then we may expand $f$ in terms of the functions $F_i$: $f(x)=\sum_{n=1}^\infty a_n F_n(x)$.]

Definition: An orthogonal set of functions is a set of functions each of which is orthogonal on a prescribed interval to all other members of the set. [That is, the set $\{F_i\}$ is is an orthogonal set if every member is orthogonal to every other member: $\langle F_i|F_j \rangle=0$ for all $i,j$ such that $i\neq j$.]

Example: The set $\{\sin nx, \cos nx\}$, on $[-\pi,\pi]$, for $n$ zero or positive integers, is an orthogonal set of functions.

Definition: An orthonormal set of functions is an orthogonal set of functions, each of which is normalised. [That is, the set $\{F_i\}$ is orthonormal if $\langle F_i|F_j \rangle=0$ for all $i\neq j$ and $\langle F_i|F_i \rangle=1$ for all $j$.]

These two equations occurs so often in discussing orthonormal functions that a special symbol has been introduced to combine them. The Kronecker delta symbol $\delta_{ij}$ has the meaning $\delta_{ij}=0$ for $i\neq j$, $\delta_{ij}=1$ for $i=j$.

The Fourier series is an expansion of a function in the orthonormal functions which are proportional to $\{\sin mx,\cos mx\}$. Let us derive their norm on the interval $[-\pi,-\pi]$. $$N(\sin mx)=\langle \sin mx|\sin mx \rangle=\int_{-\pi}^\pi \sin^2 mx dx=\dfrac{1}{m}\int_{-m\pi}^{m\pi}\sin^2 y dy=\pi.$$ Similarly, $$N(\cos mx)=\langle \cos mx|\cos mx \rangle=\dfrac{1}{m}\int_{-m\pi}^{m\pi}\cos^2 y dy=\pi.$$ Also, $$N(\sin 0x)=\langle \sin 0x|\sin 0x \rangle=\int_{-\pi}^\pi 0 dx=0$$ and $$N(\cos 0x)=\langle \cos 0x|\cos 0x \rangle=\int_{-\pi}^{\pi}1dx=2\pi.$$ Summarising the results, $$N(\sin mx)=\begin{cases} \pi\quad m\neq 0\\ 0\quad m=0 \end{cases}\\ N(\cos mx)=\begin{cases} \pi\quad m\neq 0\\ 2\pi \quad m=0 \end{cases}$$ or using the Kronecker delta, we have $$N(\sin mx)=\pi-\pi\delta_{m0}\\N(\cos mx)=\pi+\pi\delta_{m0}.$$ To calculate the expansion coefficients for an expansion in orthonormal functions, we use the set of functions $\{\dfrac{1}{\sqrt{2\pi}},\dfrac{1}{\sqrt{\pi}}\sin mx,\dfrac{1}{\sqrt{\pi}}\cos mx\}$ for $m=1,2,\cdots$, for expansion on $[-\pi,\pi]$. For these functions, the expansion coefficients will be $$a_0=\langle \dfrac{1}{\sqrt{2\pi}}|f\rangle\\a_m=\langle \dfrac{1}{\sqrt{\pi}}\cos mx|f\rangle\\b_m=\langle \dfrac{1}{\sqrt{\pi}}\sin mx|f\rangle,$$ where the expansion is $$f(x)=a_0\dfrac{1}{\sqrt{2\pi}}+\sum_{m=1}^\infty a_m(\dfrac{1}{\sqrt{\pi}}\cos mx)+\sum_{m=1}^\infty b_m(\dfrac{1}{\sqrt{\pi}}\sin mx).$$ Usually the constants are removed from the terms by explicitly writing out the expansion coefficients $$f(x)=\dfrac{1}{2\pi}\langle 1|f\rangle+\dfrac{1}{\pi}\sum_{m=1}^\infty [\langle \cos mx|f\rangle \cos mx+\langle \sin mx|f\rangle \sin mx].$$ This gives the final result $$f(x)=\dfrac{c_0}{2}+\sum_{m=1}^\infty c_m \cos mx+\sum_{m=1}^\infty d_m \sin mx\\c_m=\dfrac{1}{\pi}\langle \cos mx|f \rangle\\ d_m=\dfrac{1}{\pi}\langle \sin mx|f \rangle$$ on $[-\pi,\pi]$. This is the usual form of Fourier series. Note the lead term is divided by two because the norm of $\cos 0x$ is $2\pi$, whereas for all other values of $m$, the norm of $\cos mx$ is $\pi$.

Using the property of evenness and oddness of functions, we can make two simple extensions of the Fourier series. The Fourier expansion on $[-\pi,\pi]$ of an odd function is made up of sine terms $$f(x)=\sum_{m=1}^\infty d_m \sin mx$$ since if $f$ is odd, all inner products $\langle \cos mx|f \rangle \equiv 0$. The Fourier expansion on $[-\pi,\pi]$ of an even function is made up only of cosine terms: $$f(x)=\dfrac{c_0}{2}+\sum_{m=1}^\infty c_m \cos mx$$ since if $f$ is even, all inner products $\langle \sin mx|f \rangle \equiv 0$.

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Reference:
Mathematics for Quantum Chemistry by Jay Martin Anderson