Appendix A — Mathematical Reference

This appendix gathers the formulae, identities and unit conversions that appear throughout the book. It is intended as a lookup resource; none of the material is derived in detail here. Where a result is non-trivial, a chapter cross-reference points to the derivation.

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A.1 Physical constants

In what follows, SI denotes the International System of Units and a.u. denotes Hartree atomic units, in which \(\hbar = m_e = e = 1/(4\pi\epsilon_0) = 1\).

Constant	Symbol	SI value	Atomic units
Speed of light in vacuum	\(c\)	\(2.998 \times 10^8\) m·s\(^{-1}\)	\(137.036\ (\alpha^{-1})\)
Planck constant	\(h\)	\(6.626 \times 10^{-34}\) J·s	\(2\pi\)
Reduced Planck constant	\(\hbar\)	\(1.055 \times 10^{-34}\) J·s	\(1\)
Elementary charge	\(e\)	\(1.602 \times 10^{-19}\) C	\(1\)
Electron mass	\(m_e\)	\(9.109 \times 10^{-31}\) kg	\(1\)
Proton mass	\(m_p\)	\(1.673 \times 10^{-27}\) kg	\(1836.15\)
Boltzmann constant	\(k_B\)	\(1.381 \times 10^{-23}\) J·K\(^{-1}\)	\(3.167 \times 10^{-6}\) Ha·K\(^{-1}\)
Vacuum permittivity	\(\epsilon_0\)	\(8.854 \times 10^{-12}\) F·m\(^{-1}\)	\(1/(4\pi)\)
Avogadro number	\(N_A\)	\(6.022 \times 10^{23}\) mol\(^{-1}\)	—
Fine-structure constant	\(\alpha\)	\(7.297 \times 10^{-3}\)	\(1/c\)
Bohr radius	\(a_0\)	\(5.292 \times 10^{-11}\) m	\(1\)
Hartree energy	\(E_h\)	\(4.360 \times 10^{-18}\) J	\(1\)

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A.2 Unit conversions

Energy

From	To	Multiply by
Ha (Hartree)	eV	\(27.2114\)
Ha	kcal·mol\(^{-1}\)	\(627.509\)
Ha	kJ·mol\(^{-1}\)	\(2625.50\)
Ha	J	\(4.3597 \times 10^{-18}\)
Ha	K (via \(k_B\))	\(3.158 \times 10^{5}\)
Ha	cm\(^{-1}\)	\(2.1947 \times 10^{5}\)
eV	J	\(1.602 \times 10^{-19}\)
eV	K	\(1.160 \times 10^{4}\)
eV	cm\(^{-1}\)	\(8065.5\)
kcal·mol\(^{-1}\)	meV	\(43.36\)
kJ·mol\(^{-1}\)	meV	\(10.36\)

The conversion energy \(\leftrightarrow\) temperature is $$ E = k_B T, $$ so \(1\) Ha corresponds to a temperature of \(E_h / k_B \approx 3.158 \times 10^5\) K, and room temperature (\(T = 300\) K) corresponds to \(k_B T \approx 25.85\) meV \(\approx 0.95 \times 10^{-3}\) Ha.

Length

From	To	Multiply by
bohr (\(a_0\))	Å	\(0.52918\)
bohr	nm	\(0.052918\)
bohr	m	\(5.2918 \times 10^{-11}\)
Å	bohr	\(1.8897\)
Å	nm	\(0.1\)

Time

From	To	Multiply by
atomic time unit	s	\(2.4189 \times 10^{-17}\)
atomic time unit	fs	\(0.02419\)
ps	fs	\(10^3\)
ns	ps	\(10^3\)

The atomic unit of time is \(\hbar / E_h\), the time taken by an electron in the ground state of hydrogen (Bohr orbit) to traverse \(1\) radian of phase.

Force, pressure, dipole moment

From	To	Multiply by
Ha / bohr	eV / Å	\(51.422\)
Ha / bohr\(^3\)	GPa	\(2.942 \times 10^4\)
eV / Å\(^3\)	GPa	\(160.218\)
a.u. (dipole)	Debye	\(2.5418\)

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A.3 Linear algebra cheatsheet

Throughout this section, \(A, B, C\) are square matrices of compatible dimension; \(\mathbf{x}, \mathbf{y}\) are column vectors.

Matrix identities

\[ (AB)^\top = B^\top A^\top, \qquad (AB)^{-1} = B^{-1} A^{-1}, $$ $$ \det(AB) = \det(A)\det(B), \qquad \det(A^\top) = \det(A), $$ $$ \det(A^{-1}) = \frac{1}{\det(A)}, \qquad \det(cA) = c^n \det(A)\ \text{for}\ A \in \mathbb{R}^{n\times n}, $$ $$ \operatorname{tr}(AB) = \operatorname{tr}(BA), \qquad \operatorname{tr}(A+B) = \operatorname{tr}(A) + \operatorname{tr}(B). \]

Eigenvalues

If \(A\mathbf{v} = \lambda\mathbf{v}\):

\[ \det(A) = \prod_i \lambda_i, \qquad \operatorname{tr}(A) = \sum_i \lambda_i. \]

For a Hermitian (or symmetric real) matrix, all eigenvalues are real and eigenvectors for distinct eigenvalues are orthogonal. Any Hermitian \(A\) admits a spectral decomposition $$ A = U \Lambda U^\dagger, $$ with \(U\) unitary and \(\Lambda\) diagonal.

The Rayleigh quotient \(R_A(\mathbf{x}) = \mathbf{x}^\dagger A \mathbf{x} / \mathbf{x}^\dagger \mathbf{x}\) is bounded between \(\lambda_\text{min}\) and \(\lambda_\text{max}\), with equality on the corresponding eigenvectors. This underlies the variational principle of Chapter 4.

Matrix derivatives

For \(\mathbf{x} \in \mathbb{R}^n\) and constant \(A\):

\[ \frac{\partial}{\partial \mathbf{x}}(\mathbf{a}^\top \mathbf{x}) = \mathbf{a}, \qquad \frac{\partial}{\partial \mathbf{x}}(\mathbf{x}^\top A \mathbf{x}) = (A + A^\top)\mathbf{x}. \]

For a scalar function \(f\) of a matrix \(X\):

\[ \frac{\partial}{\partial X}\operatorname{tr}(AX) = A^\top, \qquad \frac{\partial}{\partial X}\operatorname{tr}(AX^\top) = A, $$ $$ \frac{\partial}{\partial X}\log\det X = (X^{-1})^\top. \]

Block matrix inversion

$$ \begin{pmatrix} A & B \ C & D \end{pmatrix}^{-1} = \begin{pmatrix} A^{-1} + A^{-1} B S^{-1} C A^{-1} & -A^{-1} B S^{-1} \ -S^{-1} C A^{-1} & S^{-1} \end{pmatrix}, $$ where \(S = D - C A^{-1} B\) is the Schur complement. Useful for deriving the equations of constrained optimisation and for manipulating Kohn–Sham systems with non-orthogonal bases.

The Woodbury identity

\[ (A + UCV)^{-1} = A^{-1} - A^{-1} U (C^{-1} + V A^{-1} U)^{-1} V A^{-1}. \]

Used whenever a low-rank update to a covariance or Fock matrix must be inverted cheaply.

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A.4 Vector calculus

Standard differential operators

In Cartesian coordinates, for a scalar field \(f\) and a vector field \(\mathbf{A}\):

\[ \nabla f = \left(\partial_x f,\ \partial_y f,\ \partial_z f\right), $$ $$ \nabla \cdot \mathbf{A} = \partial_x A_x + \partial_y A_y + \partial_z A_z, $$ $$ \nabla \times \mathbf{A} = \bigl( \partial_y A_z - \partial_z A_y,\ \partial_z A_x - \partial_x A_z,\ \partial_x A_y - \partial_y A_x \bigr), $$ $$ \nabla^2 f = \partial_x^2 f + \partial_y^2 f + \partial_z^2 f. \]

Identities

For any smooth \(f\) and \(\mathbf{A}\):

\[ \nabla \times (\nabla f) = \mathbf{0}, \qquad \nabla \cdot (\nabla \times \mathbf{A}) = 0. \]

For vector fields \(\mathbf{A}, \mathbf{B}\):

\[ \nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}, $$ $$ \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B}\cdot(\nabla\times\mathbf{A}) - \mathbf{A}\cdot(\nabla\times\mathbf{B}), $$ $$ \nabla(\mathbf{A}\cdot\mathbf{B}) = (\mathbf{A}\cdot\nabla)\mathbf{B} + (\mathbf{B}\cdot\nabla)\mathbf{A} + \mathbf{A}\times(\nabla\times\mathbf{B}) + \mathbf{B}\times(\nabla\times\mathbf{A}). \]

Integral theorems

Divergence (Gauss) theorem. For a vector field \(\mathbf{F}\) defined on a region \(V\) with boundary \(\partial V\): $$ \int_V \nabla \cdot \mathbf{F} \mathrm{d}V = \oint_{\partial V} \mathbf{F} \cdot \mathbf{n} \mathrm{d}S. $$

Stokes' theorem. For a vector field \(\mathbf{F}\) on a surface \(S\) with boundary \(\partial S\): $$ \int_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \mathrm{d}S = \oint_{\partial S} \mathbf{F} \cdot \mathrm{d}\boldsymbol{\ell}. $$

Green's theorem (2D special case): $$ \oint_{\partial D} (P\,\mathrm{d}x + Q\,\mathrm{d}y) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathrm{d}A. $$

Green's identities (for scalar fields \(\phi, \psi\)): $$ \int_V (\phi \nabla^2 \psi + \nabla\phi \cdot \nabla\psi) \mathrm{d}V = \oint_{\partial V} \phi\,\nabla\psi \cdot \mathbf{n} \mathrm{d}S, $$ $$ \int_V (\phi \nabla^2 \psi - \psi \nabla^2 \phi) \mathrm{d}V = \oint_{\partial V} (\phi\nabla\psi - \psi\nabla\phi)\cdot\mathbf{n} \mathrm{d}S. $$

The second of these underlies the derivation of the Hellmann–Feynman theorem and the divergence-form Poisson solver of Chapter 6.

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A.5 Common integrals

Gaussian integrals

\[ \int_{-\infty}^{\infty} e^{-\alpha x^2}\ \mathrm{d}x = \sqrt{\frac{\pi}{\alpha}}, \qquad \alpha > 0, $$ $$ \int_{-\infty}^{\infty} x^2 e^{-\alpha x^2}\ \mathrm{d}x = \frac{1}{2\alpha}\sqrt{\frac{\pi}{\alpha}}, $$ $$ \int_{-\infty}^{\infty} e^{-\alpha x^2 + \beta x}\ \mathrm{d}x = \sqrt{\frac{\pi}{\alpha}}\,\exp\!\left(\frac{\beta^2}{4\alpha}\right). \]

In \(d\) dimensions, for a positive-definite matrix \(A\): $$ \int_{\mathbb{R}^d} e^{-\tfrac12 \mathbf{x}^\top A \mathbf{x} + \mathbf{b}^\top \mathbf{x}} \mathrm{d}^d x = \frac{(2\pi)^{d/2}}{\sqrt{\det A}}\, \exp!\left(\tfrac12 \mathbf{b}^\top A^{-1} \mathbf{b}\right). $$

Exponential / gamma integrals

\[ \int_0^\infty x^{n-1} e^{-x}\,\mathrm{d}x = \Gamma(n), \quad \Gamma(n+1) = n\,\Gamma(n),\ \Gamma(1) = 1,\ \Gamma(1/2) = \sqrt{\pi}. $$ $$ \int_0^\infty x^n e^{-\alpha x}\,\mathrm{d}x = \frac{n!}{\alpha^{n+1}}, \qquad \int_0^\infty \frac{x^{s-1}}{e^x - 1}\,\mathrm{d}x = \Gamma(s)\zeta(s). \]

The last appears in the derivation of the Debye model in Chapter 8.

Useful trigonometric / spherical integrals

\[ \int_0^\pi \sin\theta\,\mathrm{d}\theta = 2, \qquad \int_0^{2\pi}\mathrm{d}\phi = 2\pi, $$ so the solid-angle integral over the sphere is $4\pi$. The spherical harmonics satisfy $$ \int_0^{2\pi}\!\!\int_0^\pi Y_\ell^{m\,*}(\theta,\phi) Y_{\ell'}^{m'}(\theta,\phi) \sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi = \delta_{\ell\ell'}\delta_{mm'}. \]

The Coulomb integral

\[ \int_{\mathbb{R}^3} \frac{e^{-2\zeta r}}{|\mathbf{r} - \mathbf{r}'|} \,\mathrm{d}^3 r = \frac{4\pi}{(2\zeta)^2} \cdot \frac{1}{r'}\left[1 - (1 + \zeta r')e^{-2\zeta r'}\right]. \]

The cornerstone evaluation underlying every Slater-type integral in quantum chemistry. Used in Chapter 4.

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A.6 Taylor series

For a sufficiently smooth \(f\) about \(x_0\):

\[ f(x) = f(x_0) + f'(x_0)(x-x_0) + \tfrac12 f''(x_0)(x-x_0)^2 + \ldots \]

Common one-variable series (\(|x|\) small):

\[ e^x = 1 + x + \tfrac{x^2}{2!} + \tfrac{x^3}{3!} + \ldots, $$ $$ \sin x = x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!} - \ldots, $$ $$ \cos x = 1 - \tfrac{x^2}{2!} + \tfrac{x^4}{4!} - \ldots, $$ $$ \ln(1+x) = x - \tfrac{x^2}{2} + \tfrac{x^3}{3} - \ldots, $$ $$ (1+x)^\alpha = 1 + \alpha x + \tfrac{\alpha(\alpha-1)}{2}x^2 + \ldots \]

Multivariate Taylor expansion to second order about \(\mathbf{x}_0\): $$ f(\mathbf{x}) \approx f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0)^\top \delta\mathbf{x} + \tfrac12 \delta\mathbf{x}^\top H \delta\mathbf{x}, $$ where \(H_{ij} = \partial_i\partial_j f(\mathbf{x}_0)\) is the Hessian. The harmonic approximation to vibrations (Chapter 8) and the second-order optimisation schemes (Chapter 6) are direct applications.

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A.7 Probability and statistics

For a random variable \(X\) with density \(p(x)\):

\[ \mathbb{E}[X] = \int x\,p(x)\,\mathrm{d}x, \qquad \mathrm{Var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2. \]

For independent \(X, Y\): $$ \mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y). $$

Gaussian density in one dimension: $$ p(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp!\left(-\frac{(x-\mu)^2}{2\sigma2}\right). $$

Multivariate Gaussian: $$ p(\mathbf{x}) = \frac{1}{\sqrt{(2\pi)^d \det \Sigma}} \exp!\left(-\tfrac12 (\mathbf{x}-\boldsymbol{\mu})^\top \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu})\right). $$

The Boltzmann distribution at temperature \(T\) for a system with energies \(\{E_i\}\): $$ p_i = \frac{e^{-\beta E_i}}{Z}, \qquad \beta = 1/(k_B T), \qquad Z = \sum_i e^{-\beta E_i}. $$

The connection to thermodynamic free energy is \(F = -k_B T \ln Z\).

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A.8 Useful inequalities

Cauchy–Schwarz. $$ |\langle \mathbf{x}, \mathbf{y}\rangle| \le |\mathbf{x}|\,|\mathbf{y}|. $$

Triangle. $$ |\mathbf{x}+\mathbf{y}| \le |\mathbf{x}| + |\mathbf{y}|. $$

Jensen. For convex \(\varphi\) and any random variable \(X\), $$ \varphi(\mathbb{E}[X]) \le \mathbb{E}[\varphi(X)]. $$

Variational (Rayleigh–Ritz). For any normalised trial state \(|\psi\rangle\) and Hermitian \(H\), $$ \langle\psi|H|\psi\rangle \ge E_0. $$

This is the principle that underwrites every variational method in this book — Hartree–Fock, DFT, configuration interaction, quantum Monte Carlo.

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The remainder of the appendix turns to operational matters: how to run these methods on real hardware (Appendix B), what the terminology means (Appendix C), where to read further (Appendix D), and a consolidated bibliography (Appendix E).