Mathematical identities cheatsheet

A single-page reference for the math that recurs across the DFT Notes. Every identity is stated in the conventions used in the chapters (atomic units, Dirac bra-ket, real spherical harmonics, the Pople `N-M₁M₂G’ basis-set notation, etc.). Each entry is cross-referenced to the chapter that uses it, so the reader can jump straight to the worked derivation.

The cheatsheet is not a substitute for the chapters themselves. It is a lookup table: when a derivation uses an identity you have forgotten, the cheatsheet tells you what the identity is and where to find the step-by-step use of it. For proofs, see the linked chapter.

Notation. All formulas are in atomic units ($\hbar = m_e = e^2 / (4\pi\varepsilon_0) = 1$) unless otherwise stated. Lengths are in Bohr ($a_0$), energies in Hartree ($E_h$), and the charge of the electron is $-1$. The notation table in chapter 00 is the source of truth; this page only collects the math.

1. Dirac notation

The chapters use the Dirac bra-ket notation of chapter 01 for every inner product and every matrix element. The same notation works in $\mathbb C^n$ (for finite-basis calculations) and in $L^2(\mathbb R^3)$ (for the continuous Hilbert space).

1.1 Inner product

\[\label{eq:ch-extra-inner} \langle u \rvert v \rangle \;\equiv\; \begin{cases} \displaystyle \sum_{i=1}^{n} u_i^* v_i & \text{in } \mathbb C^n, \\\\[6pt] \displaystyle \int u^*(\mathbf r)\, v(\mathbf r)\, d\mathbf r & \text{in } L^2(\mathbb R^3). \end{cases}\]

Used to define the Born rule, the Fock and overlap matrix elements in the AO basis $F_{\mu\nu} = \langle \chi_\mu \rvert \hat F \rvert \chi_\nu \rangle$, and the MO normalisation $\langle \phi_i \rvert \phi_j \rangle = \delta_{ij}$. Cross-reference: chapters 01 (§1.2), 03 (§3.2, §3.6).

1.2 Outer product

\[\label{eq:ch-extra-outer} \rvert u \rangle \langle v \rvert \;:\; \text{the rank-one operator that maps } \rvert w \rangle \mapsto \rvert u \rangle \langle v \rvert w \rangle .\]

Used to build the closed-shell AO density matrix as $\mathbf P = 2 \sum_i \rvert \phi_i \rangle \langle \phi_i \rvert$ and the projector onto the occupied subspace in a finite basis. Cross-reference: chapter 03 (§3.6.4, the idempotency $\mathbf P \mathbf S \mathbf P = \mathbf P$).

1.3 Completeness (resolution of the identity)

\[\label{eq:ch-extra-completeness} \hat{\mathbf 1} \;=\; \sum_{i=1}^{K} \rvert \chi_i \rangle \langle \chi_i \rvert \;\;(\text{discrete basis}), \qquad \hat{\mathbf 1} \;=\; \int \rvert \mathbf r \rangle \langle \mathbf r \rvert\, d\mathbf r \;\;(\text{position basis}).\]

Inserted between operators in derivations to convert operator equations to matrix form. The two forms above are equivalent when ${\chi_i}$ is a complete orthonormal set; for a non-orthogonal basis the form $\mathbf S^{-1} = \sum_i \rvert \chi^i \rangle \langle \chi_i \rvert$ requires the dual vectors $\rvert \chi^i \rangle$. Cross-reference: chapters 01 (§1.8.3, transition amplitudes), 06 (§6.1, Roothaan–Hall).

1.4 Projection operators

A projector $\hat P$ satisfies $\hat P^2 = \hat P$ and $\hat P^\dagger = \hat P$. The projector onto the subspace spanned by the $K$ vectors ${\rvert \chi_i \rangle}$ (in general non-orthogonal) is

\[\label{eq:ch-extra-projector} \hat P \;=\; \sum_{i,j=1}^{K} \rvert \chi_i \rangle\, (\mathbf S^{-1})_{ij}\, \langle \chi_j \rvert , \qquad \hat P^2 = \hat P, \quad \hat P^\dagger = \hat P .\]

For an orthonormal set the matrix $\mathbf S^{-1}$ reduces to $\mathbf 1$, giving $\hat P = \sum_i \rvert \chi_i \rangle \langle \chi_i \rvert$. The idempotency of $\mathbf P$ in the AO basis ($\mathbf P \mathbf S \mathbf P = \mathbf P$) is exactly the statement that the density matrix is a projector onto the occupied subspace. Cross-reference: chapter 03 (§3.6.4).

Tip. The Hermitian conjugate of $\rvert u \rangle \langle v \rvert$ is $\rvert v \rangle \langle u \rvert$. The trace of an outer product is $\operatorname{Tr}\Bigl(\rvert u \rangle \langle v \rvert\Bigr) = \langle v \rvert u \rangle$. Both facts are used in the half-trace form of the HF energy, $E_\text{HF} = \tfrac{1}{2} \operatorname{Tr}\Bigl[ \mathbf P (\mathbf h + \mathbf F) \Bigr]$.

2. Operator identities

2.1 The commutator and its algebra

\[\label{eq:ch-extra-commutator} [\hat A, \hat B] \;\equiv\; \hat A \hat B - \hat B \hat A .\]

The commutator is bilinear, antisymmetric $([\hat A, \hat B] = -[\hat B, \hat A]$), and satisfies the Jacobi identity

\[\label{eq:ch-extra-jacobi} [\hat A, [\hat B, \hat C]] + [\hat B, [\hat C, \hat A]] + [\hat C, [\hat A, \hat B]] \;=\; 0 .\]

The canonical commutation relation (CCR) of quantum mechanics,

\[\label{eq:ch-extra-ccr} [\hat r_a, \hat p_b] \;=\; i\, \delta_{ab} \qquad (a, b \in \{x, y, z\}) ,\]

defines the algebra on which the Heisenberg uncertainty principle $\Delta A \Delta B \ge \tfrac{1}{2} \lvert \langle [\hat A, \hat B] \rangle \rvert$ rests. Cross-reference: chapter 01 (§1.4, §1.7.5, problem 1 of §1.13).

2.2 The anticommutator

\[\label{eq:ch-extra-anticommutator} \{\hat A, \hat B\} \;\equiv\; \hat A \hat B + \hat B \hat A .\]

For fermionic creation and annihilation operators, ${\hat a, \hat a^\dagger} = 1$, ${\hat a, \hat a} = 0$ — the algebra of the canonical anticommutation relation (CAR). The fermionic CAR is what makes a Slater determinant antisymmetric. Cross-reference: chapter 01 (§1.2, postulate P6).

2.3 Baker–Campbell–Hausdorff (BCH)

The product of two exponentials of non-commuting operators re-exponentiates as

\[\label{eq:ch-extra-bch} e^{\hat A} e^{\hat B} \;=\; e^{\hat A + \hat B + \tfrac{1}{2}[\hat A, \hat B] + \tfrac{1}{12}[\hat A, [\hat A, \hat B]] - \tfrac{1}{12}[\hat B, [\hat A, \hat B]] + \cdots} .\]

The series terminates at finite order only when the nested commutators eventually vanish (e.g. $[\hat A, \hat B]$ is a $c$-number, as for the position-momentum pair). For $\hat A = \lambda \hat X$, $\hat B = \lambda \hat Y$ with $[\hat X, \hat Y] = \hat Z$ independent of $\hat X, \hat Y$, the BCH formula gives the Zassenhaus formula

\[\label{eq:ch-extra-zassenhaus} e^{\hat A + \hat B} \;=\; e^{\hat A}\, e^{\hat B}\, e^{-\tfrac{1}{2}[\hat A, \hat B]}\, e^{\tfrac{1}{6}(2[\hat B, [\hat A, \hat B]] + [\hat A, [\hat A, \hat B]])} \cdots ,\]

i.e. the product of two exponentials is the exponential of the sum plus correction factors. Used in the derivation of the time-evolution operator in the interaction picture (chapter 01, §1.8) and in the factorisation of the Fock matrix (chapter 03).

2.4 Useful specific identities

\(e^{\hat A} \hat B e^{-\hat A} \;=\; \hat B + [\hat A, \hat B] + \tfrac{1}{2!}[\hat A, [\hat A, \hat B]] + \tfrac{1}{3!}[\hat A, [\hat A, [\hat A, \hat B]]] + \cdots \quad \text{(similarity transform)} , \label{eq:ch-extra-sim}\) \([\hat A \hat B, \hat C] \;=\; \hat A [\hat B, \hat C] + [\hat A, \hat C] \hat B \quad \text{(product rule)} , \label{eq:ch-extra-prodcom}\) \(\operatorname{Tr}[\hat A, \hat B] \;=\; 0 \quad \text{(trace of a commutator)} . \label{eq:ch-extra-tracecom}\)

The first is the Hadamard lemma (or Campbell identity) in series form. The second lets you reduce a commutator with a product to a sum of simpler commutators. The third is the cyclicity of the trace applied to a commutator; it is the reason the Fock-operator trace formulas of chapter 03 (§3.2) collapse so cleanly. Cross-reference: chapters 01, 03. —

3. Hilbert-space identities

3.1 Trace identities

\(\operatorname{Tr}(\mathbf A \mathbf B) \;=\; \operatorname{Tr}(\mathbf B \mathbf A) , \label{eq:ch-extra-tracecyc}\) \(\operatorname{Tr}(\mathbf A + \mathbf B) \;=\; \operatorname{Tr}\mathbf A + \operatorname{Tr}\mathbf B , \label{eq:ch-extra-tracelin}\) \(\operatorname{Tr}(\mathbf A \mathbf B \mathbf C) \;=\; \operatorname{Tr}(\mathbf C \mathbf A \mathbf B) = \operatorname{Tr}(\mathbf B \mathbf C \mathbf A) \quad \text{(cyclic)} . \label{eq:ch-extra-tracecyc3}\)

The cyclic property is the engine behind every trace formula in DFT: $E = \operatorname{Tr}[\mathbf P \mathbf h]$ follows because $\mathbf P$ and $\mathbf h$ can be cyclically permuted inside the trace. Cross-reference: chapters 03 (§3.2, §3.6.5), 04 (§4.4).

3.2 Trace of a function of a matrix

If $\mathbf A$ has eigenvalues $\{ a_i \}$ and eigenvectors ${\rvert v_i \rangle}$, then for any function $f$ that admits a Taylor series,

\[\label{eq:ch-extra-tracef} \operatorname{Tr} f(\mathbf A) \;=\; \sum_{i=1}^{K} f(a_i) .\]

The same is true for $f(\hat A)$ in any finite-dimensional representation: $\operatorname{Tr} f(\hat A) = \sum_i f(a_i)$ where the sum is over the spectrum of $\hat A$. Used in the proof that the sum of orbital energies equals the trace of the Fock matrix, and in deriving the Janak theorem from the Kohn–Sham energy expression.

3.3 Hilbert–Schmidt inner product

\[\label{eq:ch-extra-hs} \langle \mathbf A, \mathbf B \rangle_\text{HS} \;\equiv\; \operatorname{Tr}(\mathbf A^\dagger \mathbf B) .\]

The Frobenius / Hilbert–Schmidt norm is $\lVert \mathbf A \rVert_\text{HS}^2 = \operatorname{Tr}(\mathbf A^\dagger \mathbf A)$. This is the inner product used in the DIIS metric $\mathbf B_{ij} = \langle R_i, R_j \rangle$ in chapter 04 (§4.6.2). Cross-reference: chapter 04. ### 3.4 Trace of a projector

\[\label{eq:ch-extra-traceproj} \operatorname{Tr} \hat P \;=\; \dim(\text{range of } \hat P) .\]

In a finite basis, $\operatorname{Tr} \mathbf P = N_\text{occ}$ (counting spins separately). Used in the constraint that the density matrix integrates to the number of electrons.

4. Second quantisation

Second quantisation replaces the antisymmetrised Slater determinants of chapter 02 (§2.2) by creation and annihilation operators acting on a Fock space. This section collects the identities the chapters use.

4.1 Bosonic creation and annihilation operators

\([\hat a, \hat a^\dagger] \;=\; \hat 1 , \label{eq:ch-extra-bosonic}\) \([\hat a, \hat a] \;=\; [\hat a^\dagger, \hat a^\dagger] \;=\; 0 . \label{eq:ch-extra-bosonic2}\)

The number operator $\hat n = \hat a^\dagger \hat a$ has eigenvalues $n = 0, 1, 2, \dots$ (any non-negative integer) and spectrum $E_n = \omega(n + 1/2)$ for the harmonic oscillator. Cross-reference: chapter 01 (§1.9, ladder operators).

4.2 Fermionic creation and annihilation operators

\(\{\hat c_p, \hat c_q^\dagger\} \;=\; \delta_{pq} , \label{eq:ch-extra-fermionic}\) \(\{\hat c_p, \hat c_q\} \;=\; \{\hat c_p^\dagger, \hat c_q^\dagger\} \;=\; 0 . \label{eq:ch-extra-fermionic2}\)

The $\delta_{pq}$ is Kronecker, not Dirac — the anticommutation is only between operators in the same single- particle basis. The CAR is what enforces the Pauli exclusion principle: $\hat c_p^\dagger \hat c_p^\dagger = 0$ means no state can be doubly created. Cross-reference: chapter 02 (§2.2).

4.3 Field operators

A general field operator is the superposition over a single-particle basis

\[\label{eq:ch-extra-field} \hat \psi(\mathbf r) \;=\; \sum_{p} \hat c_p\, \chi_p(\mathbf r) , \qquad \hat \psi^\dagger(\mathbf r) \;=\; \sum_{p} \hat c_p^\dagger\, \chi_p^*(\mathbf r) .\]

For fermions, $\hat \psi$ and $\hat \psi^\dagger$ satisfy the continuum CAR

\[\label{eq:ch-extra-fieldcar} \{ \hat\psi(\mathbf r), \hat\psi^\dagger(\mathbf r') \} \;=\; \delta(\mathbf r - \mathbf r') , \qquad \{ \hat\psi(\mathbf r), \hat\psi(\mathbf r') \} \;=\; 0 .\]

The one-body density is $\hat\rho(\mathbf r) = \hat\psi^\dagger(\mathbf r) \hat\psi(\mathbf r)$; its expectation value is the one-particle density matrix of chapter 03 (§3.6.4). Cross-reference: chapters 02, 03. ### 4.4 Wick’s theorem (statement)

For a product of $2n$ creation/annihilation operators, Wick’s theorem says

\[\label{eq:ch-extra-wick} \hat c_1 \hat c_2 \cdots \hat c_{2n} \;=\; \sum_{\text{all full pairings}} (\pm) \; \widehat{\hat c_{i_1} \hat c_{i_2}}\, \widehat{\hat c_{i_3} \hat c_{i_4}} \cdots \widehat{\hat c_{i_{2n-1}} \hat c_{i_{2n}}} ,\]

where the hat denotes a contraction (the vacuum expectation value $\langle 0 \rvert \hat c_i \hat c_j \rvert 0 \rangle$, or its generalisation to a non-vacuum reference) and the $(\pm)$ sign is the parity of the permutation needed to bring the paired operators next to each other. The generalisation to normal ordering with respect to a non-vacuum reference is the workhorse of every many-body perturbation theory (MP2, CCSD, …). Cross-reference: chapter 02 (§2.3, the hierarchy of post-HF methods).

Tip. The simplest non-trivial case is the product of two operators, $\hat c_1 \hat c_2 = :!\hat c_1 \hat c_2!:

  • \widehat{\hat c_1 \hat c_2}$, where $:\cdots:$ is the normal-ordered product (annihilators to the right) and the contraction is a $c$-number. Wick’s theorem for the 4-operator case $\hat c_1^\dagger \hat c_2 \hat c_3^\dagger \hat c_4$ is the Slater–Condon rules of chapter 03 (§3.4).

5. Fourier transforms in 1, 2, 3 dimensions

The chapters use the physics convention for the Fourier transform (no $1/(2\pi)^{n/2}$ prefactor in the forward transform, the inverse carries the prefactor). This is the convention of chapter 07 (§7.5) and the convention implicitly assumed in chapter 04 (§4.6.4, the Kerker preconditioner).

5.1 One dimension

\(\tilde f(k) \;=\; \int_{-\infty}^{\infty} f(x)\, e^{-i k x}\, dx , \label{eq:ch-extra-ft1d-fwd}\) \(f(x) \;=\; \frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde f(k)\, e^{+i k x}\, dk . \label{eq:ch-extra-ft1d-inv}\)

The Parseval / Plancherel identity is $\int |f(x)|^2\, dx = (1/2\pi) \int |\tilde f(k)|^2\, dk$. Cross-reference: chapter 01 (§1.11.3, the free-particle propagator by path integration).

5.2 Three dimensions

\(\tilde f(\mathbf k) \;=\; \int_{\mathbb R^3} f(\mathbf r)\, e^{-i \mathbf k \cdot \mathbf r} \\, d\mathbf r , \label{eq:ch-extra-ft3d-fwd}\) \(f(\mathbf r) \;=\; \frac{1}{(2\pi)^3} \int_{\mathbb R^3} \tilde f(\mathbf k)\, e^{+i \mathbf k \cdot \mathbf r} \\, d\mathbf k . \label{eq:ch-extra-ft3d-inv}\)

For a cell-periodic function $u(\mathbf r)$ on a lattice with primitive cell volume $\Omega$, the natural transform is the discrete Fourier series on the reciprocal lattice ${\mathbf G}$:

\(u(\mathbf r) \;=\; \frac{1}{\sqrt{\Omega}} \sum_{\mathbf G} \tilde u(\mathbf G)\, e^{i \mathbf G \cdot \mathbf r} , \label{eq:ch-extra-fs-fwd}\) \(\tilde u(\mathbf G) \;=\; \frac{1}{\sqrt{\Omega}} \int_\Omega u(\mathbf r)\, e^{-i \mathbf G \cdot \mathbf r} \\, d\mathbf r . \label{eq:ch-extra-fs-inv}\)

The reciprocal-lattice vectors are defined by $\mathbf a_i \cdot \mathbf b_j = 2\pi \delta_{ij}$ (chapter 07, §7.4.1). The plane-wave basis function $\Omega^{-1/2} e^{i(\mathbf k + \mathbf G) \cdot \mathbf r}$ of chapter 07 (§7.5) and chapter 06 (§6.7) uses this convention.

5.3 The Fourier transform of $1/r$ and $1/r^2$

The two transforms every DFT code needs:

\(\int_{\mathbb R^3} \frac{e^{-i \mathbf k \cdot \mathbf r}}{r}\, d\mathbf r \;=\; \frac{4\pi}{k^2} , \label{eq:ch-extra-fourier-1r}\) \(\int_{\mathbb R^3} \frac{e^{-i \mathbf k \cdot \mathbf r}}{r^2}\, d\mathbf r \;=\; \frac{\pi^2}{k} \quad \text{(distributional sense)} . \label{eq:ch-extra-fourier-1r2}\)

The first is the Coulomb kernel in reciprocal space — the Fourier transform of $1/r$ — and is the foundation of the Ewald-summed Coulomb interaction in periodic codes (chapter 07). Cross-reference: chapter 04 (§4.4, the Hartree operator in reciprocal space), §13 below.

5.4 Useful transforms on a finite interval

The sinc kernel

\[\label{eq:ch-extra-sinc} \int_{-L/2}^{L/2} e^{-i k x}\, dx \;=\; L\, \frac{\sin(kL/2)}{kL/2} \;=\; L\, \operatorname{sinc}(kL/2)\]

arises whenever a function is truncated to a box of length $L$. In 3-D it generalises to a product of three 1-D sincs. Used in the planar-averaging and slab-coupling corrections of chapter 07. —

6. Green’s functions

Green’s functions are the second-quantised, frequency-dependent generalisation of the resolvent $(\omega - \hat H)^{-1}$. Every spectral property of a Hamiltonian — density of states, spectral function, optical conductivity — is a Green’s function in disguise. The chapters use the imaginary-frequency / real- time conventions of chapter 04 (the Lindhard function in §4.6.4) and chapter 07 (the smearing Fermi function in §7.6.3).

6.1 The resolvent and the free-particle Green’s function

The resolvent of a Hamiltonian $\hat H$ at complex frequency $z$ is

\[\label{eq:ch-extra-resolvent} \hat G(z) \;\equiv\; (z - \hat H)^{-1} .\]

For the free-particle Hamiltonian $\hat H_0 = -\tfrac{1}{2} \nabla^2$, the resolvent in position space is

\[\label{eq:ch-extra-g0} G_0(\mathbf r, \mathbf r'; E) \;=\; -\frac{e^{i k \lvert \mathbf r - \mathbf r' \rvert}}{2\pi \lvert \mathbf r - \mathbf r' \rvert} , \qquad k = \sqrt{2E} ,\]

with the $i\eta$ prescription $E \to E + i\eta$ fixing the boundary condition (outgoing waves for $E > 0$, decaying exponential for $E < 0$). Used in the Born series for scattering, the Lippmann–Schwinger equation, and the GW self-energy of many-body perturbation theory. Cross- reference: chapters 01 (§1.8, the time-domain Dyson series), 03 (the Fock-operator resolvent).

6.2 The Matsubara / imaginary-frequency form

For a finite-temperature Matsubara Green function at the fermionic frequencies $i\omega_n = i(2n+1)\pi / \beta$ (with $\beta = 1/k_B T$),

\[\label{eq:ch-extra-matsubara} \mathcal G(\mathbf r, \mathbf r'; i\omega_n) \;=\; \int_0^\beta d\tau\, e^{i\omega_n \tau}\, \Bigl\langle \hat\psi(\mathbf r, \tau)\, \hat\psi^\dagger(\mathbf r', 0) \bigr\rangle ,\]

where $\hat\psi(\tau) = e^{\tau \hat H}\, \hat\psi\, e^{-\tau \hat H}$ is the Heisenberg-picture field. The Matsubara sum is the finite-temperature version of the contour-ordered Green function.

6.3 The Dyson equation

If the full Hamiltonian is $\hat H = \hat H_0 + \hat V$, the Dyson equation relates the full Green function $\hat G$ to the free one $\hat G_0$ and the self-energy $\hat\Sigma$:

\[\label{eq:ch-extra-dyson} \hat G(z) \;=\; \hat G_0(z) \;+\; \hat G_0(z)\, \hat\Sigma(z)\, \hat G(z) .\]

Iterating gives the Born series $\hat G = \hat G_0 + \hat G_0 \hat V \hat G_0 + \hat G_0 \hat V \hat G_0 \hat V \hat G_0 + \cdots$. In the time domain, the Dyson equation is the Dyson series of chapter 01 (§1.8.2). Cross-reference: chapter 01 (§1.8), chapter 04 (§4.6.4, the Lindhard response function as a Green-function product).

6.4 The spectral function

The spectral function is the discontinuity of $\hat G$ across the real axis,

\[\label{eq:ch-extra-spectral} \mathbf A(\omega) \;\equiv\; \frac{i}{2\pi} \Bigl[ \hat G(\omega + i\eta) - \hat G(\omega - i\eta) \Bigr] ,\]

and is positive semi-definite with unit weight $\int \mathbf A(\omega)\, d\omega = \hat 1$. In the Lehmann representation of a many-body Hamiltonian,

\[\label{eq:ch-extra-lehmann} A_{ij}(\omega) \;=\; \sum_{n} \Bigl[ \langle i \rvert n \rangle \langle n \rvert j \rangle \Bigr]\, \delta(\omega - (E_n - E_0)) ,\]

where the sum runs over the many-body eigenstates $|n\rangle$. The spectral function is the bridge from the Green-function language of many-body theory to the band-structure language of chapter 07: at the Hartree–Fock level ($\hat\Sigma = \hat V_\text{HF}$), $A(\omega)$ is a sum of delta peaks at the HF orbital energies, weighted by the spectral weights of the one-particle states. Cross- reference: chapters 01, 03, 04, 07. > Tip. The density of states is

$g(\omega) = \operatorname{Tr} \mathbf A(\omega) = \sum_i A_{ii}(\omega)$. In Kohn–Sham DFT the spectral function is approximated by the KS density of states, which is a sum of delta peaks at the KS eigenvalues — this is the physical content of the “Kohn–Sham band structure” of chapter 07. —

7. Special functions

7.1 Hermite polynomials

The physicists’ Hermite polynomials are defined by

\[\label{eq:ch-extra-hermite} H_n(x) \;=\; (-1)^n e^{x^2}\, \frac{d^n}{dx^n}\, e^{-x^2} .\]

The first few are $H_0 = 1$, $H_1 = 2x$, $H_2 = 4x^2 - 2$, $H_3 = 8x^3 - 12x$, $H_4 = 16x^4 - 48x^2 + 12$. They satisfy the orthogonality relation

\[\label{eq:ch-extra-hermite-orth} \int_{-\infty}^{\infty} H_m(x)\, H_n(x)\, e^{-x^2}\, dx \;=\; 2^n n!\,\sqrt{\pi}\, \delta_{mn} .\]

The generating function $\exp(2xt - t^2) = \sum_n H_n(x) t^n / n!$ is the link to the Glauber coherent states of chapter 01 (§1.9.5). Cross- reference: chapter 01 (§1.9.4), chapter 06 (§6.13.5, the NAO polarisation-orbital recipe uses energy derivatives of bound- state radial functions, which in the harmonic-oscillator basis are Hermite polynomials).

7.2 Laguerre polynomials

The associated Laguerre polynomials $L_n^\alpha(x)$ are defined by the generating function

\[\label{eq:ch-extra-laguerre-gen} \frac{e^{-x t/(1-t)}}{(1-t)^{\alpha+1}} \;=\; \sum_{n=0}^{\infty} L_n^\alpha(x)\, t^n ,\]

or, equivalently, by the differential equation

\[\label{eq:ch-extra-laguerre-eq} x\, \frac{d^2 L_n^\alpha}{dx^2} + (\alpha + 1 - x)\, \frac{dL_n^\alpha}{dx} + n\, L_n^\alpha(x) \;=\; 0 .\]

The orthogonality relation is

\[\label{eq:ch-extra-laguerre-orth} \int_0^\infty L_m^\alpha(x)\, L_n^\alpha(x)\, x^\alpha e^{-x}\, dx \;=\; \frac{\Gamma(n+\alpha+1)}{n!}\, \delta_{mn} .\]

The hydrogen radial wavefunctions of chapter 01 (§1.10) are $L_{n-\ell-1}^{2\ell+1}$ evaluated at $2Zr/n$. Cross-reference: chapter 01 (§1.10.5), chapter 06 (the radial STO primitives).

7.3 Legendre polynomials

The Legendre polynomials $P_\ell(\cos\theta)$ are restrictions of the spherical harmonics to $m = 0$. They satisfy the addition theorem

\[\label{eq:ch-extra-legen-add} P_\ell(\cos\gamma) \;=\; \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} Y_\ell^{m*}(\hat{\mathbf n}_1)\, Y_\ell^m(\hat{\mathbf n}_2) ,\]

where $\gamma$ is the angle between the two unit vectors $\hat{\mathbf n}_1$ and $\hat{\mathbf n}_2$. This identity is the workhorse of the molecular multipole expansion and of the angular-momentum coupling that underlies the Gaunt integral of chapter 01 (§1.10.7). Cross-reference: chapter 01 (§1.10.7).

7.4 Spherical harmonics

The spherical harmonics are the simultaneous eigenfunctions of $\hat L^2$ and $\hat L_z$ in $L^2(S^2)$:

\[\label{eq:ch-extra-sph} \hat L^2 Y_\ell^m(\theta, \phi) \;=\; \ell(\ell+1)\, Y_\ell^m , \qquad \hat L_z Y_\ell^m \;=\; m\, Y_\ell^m ,\]

with $\ell = 0, 1, 2, \dots$ and $-\ell \le m \le \ell$. The orthonormality is

\[\label{eq:ch-extra-sph-orth} \int_0^{2\pi} d\phi \int_0^{\pi} \sin\theta\, d\theta\, Y_\ell^{m*}(\theta, \phi)\, Y_{\ell'}^{m'}(\theta, \phi) \;=\; \delta_{\ell\ell'}\, \delta_{mm'} .\]

The chapters use the Condon–Shortley phase convention (an explicit $(-1)^m$ for $m > 0$), and the real spherical harmonics for cubic harmonics. The parity $Y_\ell^m(-\hat{\mathbf r}) = (-1)^\ell Y_\ell^m(\hat{\mathbf r})$ is what gives the selection rule $\Delta \ell = \pm 1$ for electric-dipole transitions (chapter 01, §1.10.8). Cross-reference: chapters 01 (§1.10.2, §1.10.7), 03 (the Gaunt integral), 06 (real spherical harmonics in NAO basis sets).

7.5 Bessel functions

The Bessel functions of the first kind $J_\nu(x)$ satisfy Bessel’s equation

\[\label{eq:ch-extra-bessel} x^2 \frac{d^2 J_\nu}{dx^2} + x \frac{dJ_\nu}{dx} + (x^2 - \nu^2)\, J_\nu \;=\; 0 ,\]

with the small- and large-$x$ asymptotics $J_\nu(x) \sim (x/2)^\nu / \Gamma(\nu+1)$ for $x \to 0$ and $J_\nu(x) \sim \sqrt{2/(\pi x)} \cos(x - \nu\pi/2 - \pi/4)$ for $x \to \infty$. The spherical Bessel functions $j_\ell(x) = \sqrt{\pi/(2x)}\, J_{\ell+1/2}(x)$ are the radial parts of the free-particle spherical waves. The free-particle Green function of equation \eqref{eq:ch-extra-g0} admits the partial-wave expansion

\[\label{eq:ch-extra-g0-pw} G_0(\mathbf r, \mathbf r'; E) \;=\; -i k \sum_{\ell, m} j_\ell(k r_<)\, h^{(1)}_\ell(k r_>)\, Y_\ell^m(\hat{\mathbf r})\, Y_\ell^{m*}(\hat{\mathbf r}') ,\]

where $h^{(1)}\ell$ is the spherical Hankel function of the first kind and $r< = \min(r, r’)$, $r_> = \max(r, r’)$. Cross-reference: chapter 01 (§1.11.3, free-particle path integral), chapter 07 (Bessel-function solution of the nearly-free-electron band structure).

7.6 The gamma function

The Euler gamma function

\[\label{eq:ch-extra-gamma} \Gamma(z) \;=\; \int_0^\infty t^{z-1} e^{-t}\, dt , \qquad \operatorname{Re} z > 0 ,\]

extends the factorial to non-integer arguments: $\Gamma(n) = (n-1)!$ for positive integer $n$. The reflection formula

\[\label{eq:ch-extra-gamma-reflect} \Gamma(z)\, \Gamma(1-z) \;=\; \frac{\pi}{\sin(\pi z)} ,\]

and the Legendre duplication formula

\[\label{eq:ch-extra-gamma-dup} \Gamma(z)\, \Gamma\!\left(z + \tfrac{1}{2}\right) \;=\; \frac{\sqrt{\pi}}{2^{2z-1}}\, \Gamma(2z) ,\]

are the identities used in evaluating the Gaussian overlaps and Boys-function integrals of chapter 06. The Bohr radius and Hartree energy of atomic units are $\Gamma$-function normalisations in disguise. Cross-reference: chapters 01, 03, 06. ### 7.7 The Boys function

The Boys function

\[\label{eq:ch-extra-boys} F_0(t) \;=\; \int_0^1 e^{-t u^2}\, du \;=\; \frac{1}{2}\sqrt{\frac{\pi}{t}}\, \operatorname{erf}(\sqrt{t}) , \qquad F_0(0) = 1 ,\]

handles the Coulomb singularity $1/r_{12}$ inside the primitive four-centre electron-repulsion integral of chapter 06 (§6.3). The general $F_n(t) = \int_0^1 u^{2n} e^{-t u^2}\, du$ appears in the higher-angular-momentum recursion. Cross-reference: chapter 06 (§6.3), chapter 03 (§3.6.3).

7.8 The error function

The error function $\operatorname{erf}(x)$ and its complement $\operatorname{erfc}(x) = 1 - \operatorname{erf}(x)$:

\[\label{eq:ch-extra-erf} \operatorname{erf}(x) \;=\; \frac{2}{\sqrt\pi} \int_0^x e^{-t^2}\, dt , \qquad \operatorname{erfc}(x) \;=\; \frac{2}{\sqrt\pi} \int_x^\infty e^{-t^2}\, dt .\]

Asymptotics: $\operatorname{erf}(x) \to 1$ as $x \to \infty$ and $\operatorname{erfc}(x) \sim e^{-x^2}/(x\sqrt\pi)$ as $x \to \infty$. Used in the Boys function above, in the range-separation of the Coulomb kernel (chapter 05, §5.5), and in the Gaussian smearing occupation function (chapter 07, §7.6.3). Cross-reference: chapters 03, 05, 06, 07. —

8. Tensor identities

8.1 The Kronecker delta

\[\label{eq:ch-extra-kron} \delta_{ij} \;=\; \begin{cases} 1, & i = j, \\\\ 0, & i \ne j. \end{cases}\]

The completeness of an orthonormal basis is $\sum_i \lvert i \rangle \langle i \rvert = \hat 1$, which in index notation is $\sum_i v_i w_i = \mathbf v \cdot \mathbf w$ and in tensor notation is $\sum_i e_i \otimes e_i = \mathbf 1$. The Kronecker $\delta$ is the identity tensor on $\mathbb R^n$.

8.2 The Levi–Civita symbol

The three-dimensional Levi–Civita symbol is totally antisymmetric in its three indices:

\[\label{eq:ch-extra-levicivita} \epsilon_{ijk} \;=\; \begin{cases} +1 & \text{even permutation of } (1, 2, 3), \\\ -1 & \text{odd permutation of } (1, 2, 3), \\\ 0 & \text{otherwise}. \end{cases}\]

The two Levi–Civita contraction identities every DFT code uses are

\(\sum_k \epsilon_{ijk}\, \epsilon_{\ell m k} \;=\; \delta_{i\ell}\, \delta_{jm} - \delta_{im}\, \delta_{j\ell} , \label{eq:ch-extra-eps2}\) \(\sum_j \epsilon_{ijk}\, \epsilon_{\ell j m} \;=\; \delta_{i\ell}\, \delta_{km} - \delta_{im}\, \delta_{k\ell} . \label{eq:ch-extra-eps1}\)

The cross product is $(\mathbf a \times \mathbf b)i = \sum{jk} \epsilon_{ijk}\, a_j b_k$; the scalar triple product is $\mathbf a \cdot (\mathbf b \times \mathbf c) = \sum_{ijk} \epsilon_{ijk}\, a_i b_j c_k$. The angular momentum $\hat{\mathbf L} = \hat{\mathbf r} \times \hat{\mathbf p}$ is the prototype.

8.3 Einstein summation convention

Repeated upper and lower indices are summed over their range. For an orthogonal basis, upper and lower indices are equivalent and the convention reduces to summing over any repeated index. Examples used in the chapters:

\(\mathbf F \mathbf c_i \;=\; \sum_{\mu\nu} F_{\mu\nu}\, c_{\nu i} \quad \text{(matrix–vector)} , \label{eq:ch-extra-eins-mat}\) \(E_\text{el} \;=\; \frac{1}{2} \sum_{\mu\nu} P_{\nu\mu}\, (h_{\mu\nu} + F_{\mu\nu}) \quad \text{(half-trace)} . \label{eq:ch-extra-eins-half}\)

The `numpy.einsum’ call in chapter 03 (§3.3) implements these contractions verbatim. Cross-reference: chapters 03, 04, 06. ### 8.4 Rotation of tensors

Under an orthogonal transformation $R$ with $R^T R = \mathbf 1$ and $\det R = +1$ (proper rotation), a rank-2 tensor transforms as $A’{ij} = \sum{k\ell} R_{ik}\, R_{j\ell}\, A_{k\ell}$. For a vector ($\mathbf v’ = R \mathbf v$) and a scalar ($\mathbf v^T A \mathbf w$ invariant), the pattern is the same. The Wigner D-matrix $D^\ell_{m,m’}(R)$ is the $\ell$-dimensional representation of $R$ on the $Y_\ell^m$ basis. Cross-reference: chapter 07 (the action of the point group on Bloch states at $\mathbf k$).

9. Linear algebra

9.1 Eigenvalues, eigenvectors, spectral theorem

A square matrix $\mathbf A$ has the eigendecomposition

\[\label{eq:ch-extra-eig} \mathbf A \mathbf v_i \;=\; a_i \mathbf v_i ,\]

where $a_i$ are the eigenvalues and $\mathbf v_i$ the eigenvectors. The spectral theorem says that a Hermitian matrix ($\mathbf A^\dagger = \mathbf A$) has real eigenvalues and a complete orthonormal set of eigenvectors. In matrix form,

\[\label{eq:ch-extra-spec} \mathbf A \;=\; \mathbf U\, \boldsymbol\Lambda\, \mathbf U^\dagger ,\]

with $\mathbf U$ unitary and $\boldsymbol\Lambda$ real diagonal. This is the diagonalisation that `numpy.linalg.eigh’ performs. The Hermitian eigenvalue problem is the workhorse of DFT — every Fock-matrix diagonalisation (HF, KS) is a Hermitian eigenproblem. Cross-reference: chapters 03 (§3.6.2), 04, 06 (§6.1).

9.2 Generalised eigenvalue problem

A generalised eigenproblem is $\mathbf A \mathbf v = \lambda \mathbf B \mathbf v$ with $\mathbf B$ positive-definite Hermitian. It reduces to the standard eigenproblem by Löwdin orthogonalisation $\mathbf X = \mathbf B^{-1/2}$:

\[\label{eq:ch-extra-gep} \mathbf X^\dagger \mathbf A \mathbf X\, \mathbf c' \;=\; \lambda\, \mathbf c', \qquad \mathbf v \;=\; \mathbf X \mathbf c' .\]

This is the Roothaan–Hall equation $\mathbf F \mathbf C = \mathbf S \mathbf C \boldsymbol\varepsilon$ of chapters 03 (§3.6) and 06 (§6.1). The standard numerical recipe is ‘scipy.linalg.eigh(F, S)’, which forms $\mathbf X$ by Cholesky factorisation of $\mathbf S$. Cross- reference: chapters 03, 06. ### 9.3 Singular value decomposition (SVD)

Every $m \times n$ matrix $\mathbf A$ admits the decomposition

\[\label{eq:ch-extra-svd} \mathbf A \;=\; \mathbf U\, \boldsymbol\Sigma\, \mathbf V^\dagger ,\]

with $\mathbf U$ $(m \times m)$ unitary, $\mathbf V$ $(n \times n)$ unitary, and $\boldsymbol\Sigma = \operatorname{diag}(\sigma_1, \dots, \sigma_r, 0, \dots)$ with $\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_r > 0$ the singular values. The SVD is the basis of the density- fitting / resolution-of-the-identity approximation to the ERI tensor of chapter 06 (§6.12). Cross-reference: chapter 06 (§6.12, the resolution-of-the-identity approximation).

9.4 Trace of a matrix

\[\label{eq:ch-extra-tr} \operatorname{Tr}\mathbf A \;=\; \sum_i A_{ii} \;=\; \sum_i a_i ,\]

where $a_i$ are the eigenvalues of $\mathbf A$. The second equality is the trace-invariance under similarity: the trace of $\mathbf A$ equals the trace of $\mathbf U^\dagger \mathbf A \mathbf U$, and the diagonal form has the eigenvalues on the diagonal. Cross-reference: chapters 03, 04. ### 9.5 Determinant of a matrix

\[\label{eq:ch-extra-det} \det \mathbf A \;=\; \prod_i a_i \;=\; e^{\operatorname{Tr}\ln \mathbf A} .\]

The last form is the Jacobi formula for the derivative of a determinant: $d(\ln \det \mathbf A) = \operatorname{Tr}(\mathbf A^{-1}\, d\mathbf A)$. The determinant of an antisymmetric matrix is the square of the Pfaffian; the determinant of a Slater matrix is the Slater determinant of chapter 02 (§2.2). Cross-reference: chapter 02. ### 9.6 Matrix exponential

The matrix exponential is defined by the Taylor series

\[\label{eq:ch-extra-matexp} e^{\mathbf A} \;=\; \sum_{n=0}^{\infty} \frac{\mathbf A^n}{n!} ,\]

which converges for every square matrix. For Hermitian $\mathbf A = \mathbf U \boldsymbol\Lambda \mathbf U^\dagger$,

\[\label{eq:ch-extra-matexp-diag} e^{\mathbf A} \;=\; \mathbf U\, e^{\boldsymbol\Lambda}\, \mathbf U^\dagger , \qquad e^{\boldsymbol\Lambda} = \operatorname{diag}(e^{\lambda_1}, e^{\lambda_2}, \dots) .\]

Used in the time-evolution operator $\hat U(t) = e^{-i \hat H t}$ of chapter 01 (§1.7.1) and in the Löwdin orthogonaliser $\mathbf X = \mathbf S^{-1/2}$ of chapter 03 (§3.6.6). Cross- reference: chapters 01, 03. ### 9.7 Sherman–Morrison–Woodbury identities

\((\mathbf A + \mathbf u \mathbf v^T)^{-1} \;=\; \mathbf A^{-1} - \frac{\mathbf A^{-1} \mathbf u \mathbf v^T \mathbf A^{-1}}{1 + \mathbf v^T \mathbf A^{-1} \mathbf u} \quad \text{(rank-1)} , \label{eq:ch-extra-sm1}\) \((\mathbf A + \mathbf U \mathbf C \mathbf V^T)^{-1} \;=\; \mathbf A^{-1} - \mathbf A^{-1} \mathbf U (\mathbf C^{-1} + \mathbf V^T \mathbf A^{-1} \mathbf U)^{-1} \mathbf V^T \mathbf A^{-1} \quad \text{(rank-}k\text{)} . \label{eq:ch-extra-smk}\)

The first is the Sherman–Morrison formula, the second the Woodbury identity. The latter is the workhorse of the Broyden update in chapter 04 (§4.6.3). Cross-reference: chapter 04. —

10. Differential operators

The chapters use the standard $\nabla$ operators of vector calculus in three coordinate systems: Cartesian, spherical polar, and cylindrical.

10.1 Cartesian

\(\nabla f \;=\; \left( \frac{\partial f}{\partial x},\, \frac{\partial f}{\partial y},\, \frac{\partial f}{\partial z} \right) , \label{eq:ch-extra-grad}\) \(\nabla \cdot \mathbf F \;=\; \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} , \label{eq:ch-extra-div}\) \(\nabla \times \mathbf F \;=\; \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z},\; \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x},\; \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) , \label{eq:ch-extra-curl}\) \(\nabla^2 f \;=\; \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} . \label{eq:ch-extra-lap-cart}\)

The vector Laplacian acts component-wise on a vector field: $\nabla^2 \mathbf F = (\nabla^2 F_x, \nabla^2 F_y, \nabla^2 F_z)$. The Laplacian on a tensor is the same: one component at a time. The gradient of a dot product is

\[\label{eq:ch-extra-grad-dot} \nabla (\mathbf A \cdot \mathbf B) \;=\; (\mathbf A \cdot \nabla) \mathbf B + (\mathbf B \cdot \nabla) \mathbf A + \mathbf A \times (\nabla \times \mathbf B) + \mathbf B \times (\nabla \times \mathbf A) .\]

10.2 Spherical polar coordinates

With $r = |\mathbf r|$, $\theta$ the polar angle from $\hat z$ and $\phi$ the azimuth,

\(\nabla f \;=\; \hat{\mathbf r}\, \frac{\partial f}{\partial r} + \hat{\boldsymbol\theta}\, \frac{1}{r}\, \frac{\partial f}{\partial \theta} + \hat{\boldsymbol\phi}\, \frac{1}{r \sin\theta}\, \frac{\partial f}{\partial \phi} , \label{eq:ch-extra-grad-sph}\) \(\nabla \cdot \mathbf F \;=\; \frac{1}{r^2} \frac{\partial (r^2 F_r)}{\partial r} + \frac{1}{r \sin\theta} \frac{\partial (\sin\theta\, F_\theta)}{\partial \theta} + \frac{1}{r \sin\theta} \frac{\partial F_\phi}{\partial \phi} , \label{eq:ch-extra-div-sph}\) \(\nabla^2 f \;=\; \frac{1}{r^2} \frac{\partial}{\partial r} \!\left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \!\left( \sin\theta\, \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 f}{\partial \phi^2} . \label{eq:ch-extra-lap-sph}\)

The angular part of $\nabla^2$ is $-\hat L^2 / r^2$, so $\nabla^2 (R Y_\ell^m) = [r^{-2} \partial_r(r^2 \partial_r R) - \ell(\ell+1)/r^2 R]\, Y_\ell^m$ — the radial equation of chapter 01 (§1.10.3). Cross-reference: chapter 01 (§1.10).

10.3 Cylindrical coordinates

With $\rho = \sqrt{x^2 + y^2}$, $\phi$ the azimuth, and $z$ the axial coordinate,

\(\nabla f \;=\; \hat{\boldsymbol\rho}\, \frac{\partial f}{\partial \rho} + \hat{\boldsymbol\phi}\, \frac{1}{\rho}\, \frac{\partial f}{\partial \phi} + \hat{\mathbf z}\, \frac{\partial f}{\partial z} , \label{eq:ch-extra-grad-cyl}\) \(\nabla^2 f \;=\; \frac{1}{\rho} \frac{\partial}{\partial \rho} \!\left( \rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} . \label{eq:ch-extra-lap-cyl}\)

The cylindrical Laplacian is the natural operator for systems with axial symmetry: nanotubes, wires, the near-degeneracy of the $p_x, p_y$ orbitals in transition-metal complexes. Used in chapter 07 when working out the band structure of a single-wall nanotube or a 2-D electron gas confined to a wire.

10.4 Vector identities

\(\nabla \cdot (\nabla \times \mathbf F) \;=\; 0 , \label{eq:ch-extra-vec-id1}\) \(\nabla \times (\nabla f) \;=\; \mathbf 0 , \label{eq:ch-extra-vec-id2}\) \(\nabla \times (\nabla \times \mathbf F) \;=\; \nabla(\nabla \cdot \mathbf F) - \nabla^2 \mathbf F , \label{eq:ch-extra-vec-id3}\) \(\nabla \cdot (f \mathbf F) \;=\; f\, \nabla \cdot \mathbf F + \mathbf F \cdot \nabla f , \label{eq:ch-extra-vec-id4}\) \(\nabla \times (f \mathbf F) \;=\; f\, \nabla \times \mathbf F + \nabla f \times \mathbf F . \label{eq:ch-extra-vec-id5}\)

Identity \eqref{eq:ch-extra-vec-id3} is the reason the Coulomb gauge $\nabla \cdot \mathbf A = 0$ and the continuity equation of chapter 01 (§1.7.4) have the structure they do. Cross-reference: chapter 01. —

11. Functional derivatives

The chapters use the functional derivative to derive the Kohn–Sham potential $\hat v_\text{xc} = \delta E_\text{xc}/\delta \rho$ (chapter 04, §4.2) and the forces on the nuclei via the Hellmann–Feynman theorem (chapter 04, §4.7.1). This section collects the working rules.

11.1 Definition

The functional derivative $\delta F / \delta f(x)$ of a functional $F[f]$ is the function (or distribution) such that for every variation $\delta f(x)$,

\[\label{eq:ch-extra-fdef} \delta F \;=\; \int \frac{\delta F}{\delta f(x)}\, \delta f(x)\, dx .\]

The simplest case: for $F[f] = \int g(x)\, f(x)\, dx$, the functional derivative is $\delta F / \delta f(x) = g(x)$. For $F[f] = f(x_0)^2$, the derivative is $\delta F / \delta f(x) = 2 f(x_0) \delta(x - x_0)$.

11.2 Chain rule

If $F = G[f, h]$ depends on $f$ and $h$ (which themselves depend on the same underlying variables), then

\[\label{eq:ch-extra-fchain} \frac{\delta F}{\delta g(y)} \;=\; \int dx\, \frac{\delta F}{\delta f(x)} \frac{\delta f(x)}{\delta g(y)} + \int dx\, \frac{\delta F}{\delta h(x)} \frac{\delta h(x)}{\delta g(y)} .\]

Used in deriving the chain rule for the KS potential: the density depends on the orbitals, the orbitals depend on the potential, and the potential depends on the density.

11.3 Integration by parts

Under a functional integral, integration by parts swaps a derivative at no cost (boundary terms vanish under the standard assumptions of the variational principle):

\[\label{eq:ch-extra-fibp} \int \frac{\delta F}{\delta f(x)} \frac{\partial f}{\partial x}\, dx \;=\; -\int f(x)\, \frac{\partial}{\partial x} \frac{\delta F}{\delta f(x)}\, dx .\]

The boundary term is $\left[ f\, \delta F/\delta f \right]_{-\infty}^{+\infty}$ and vanishes if $f \to 0$ at infinity (or if the boundary variations are restricted to vanish). Used in the derivation of the Euler–Lagrange equation in §12 below and in the Pulay-force integration by parts of chapter 04 (§4.7.3).

11.4 Functional derivative of a density-functional

For the Kohn–Sham energy $E[\rho] = T_s[\rho] + \int \rho v_\text{ext}\, d\mathbf r + J[\rho] + E_\text{xc}[\rho]$, the functional derivative with respect to $\rho(\mathbf r)$ is

\[\label{eq:ch-extra-vxc} \frac{\delta E}{\delta \rho(\mathbf r)} \;=\; v_\text{ext}(\mathbf r) + v_\text{H}[\rho](\mathbf r) + \frac{\delta E_\text{xc}}{\delta \rho(\mathbf r)} \;=\; v_\text{eff}(\mathbf r) .\]

The right-hand side is the Kohn–Sham effective potential. The functional derivative $\delta E_\text{xc}/\delta\rho$ is the XC potential of chapter 04 (§4.2, §4.5). Cross- reference: chapter 04. —

12. Variational calculus

12.1 The action

The action is the time-integral of a Lagrangian:

\[\label{eq:ch-extra-action} \mathcal S[q] \;=\; \int_{t_1}^{t_2} \mathcal L(q(t), \dot q(t), t)\, dt .\]

The principle of stationary action says the physical trajectory $q(t)$ is an extremum of $\mathcal S[q]$ against variations $\delta q(t)$ that vanish at the endpoints. The Lagrangian of the TDSE derivation of chapter 01 (§1.7.2) is the Dirac–Frenkel (McLachlan) Lagrangian $\mathcal L = \langle \psi \rvert i \partial_t - \hat H \rvert \psi \rangle$, which is a functional of the time-dependent state.

12.2 The Euler–Lagrange equation

For a Lagrangian $\mathcal L(q, \dot q, t)$, the stationary- action condition $\delta \mathcal S = 0$ gives the Euler– Lagrange equation

\[\label{eq:ch-extra-el} \frac{d}{dt} \frac{\partial \mathcal L}{\partial \dot q} - \frac{\partial \mathcal L}{\partial q} \;=\; 0 .\]

The generalisation to several coordinates is $\partial_t (\partial \mathcal L / \partial \dot q_i) - \partial \mathcal L / \partial q_i = 0$. The Euler–Lagrange equation of the Dirac–Frenkel Lagrangian of chapter 01 (§1.7.2) is the time-dependent Schrödinger equation — the TDSE is the unique dynamics consistent with stationary action and unitarity.

12.3 The Euler–Lagrange equation in field theory

For a Lagrangian density $\mathcal L = \mathcal L(\phi, \partial_\mu \phi)$,

\[\label{eq:ch-extra-el-field} \partial_\mu \frac{\partial \mathcal L}{\partial (\partial_\mu \phi)} - \frac{\partial \mathcal L}{\partial \phi} \;=\; 0 .\]

This is the Dirac–Frenkel variational principle of chapter 01 (§1.7.2) written for the wavefunction field. The time-dependent KS equations of chapter 04 are recovered by restricting $|\psi(t)\rangle$ to a single Slater determinant and varying over the orbital rotations.

12.4 Constrained variation

A constrained minimisation (e.g. orthonormality of the orbitals in HF/KS) introduces a Lagrange multiplier matrix $\boldsymbol\Lambda$ in the variational principle:

\[\label{eq:ch-extra-constraint} \delta \Bigl[ E - \operatorname{Tr}(\boldsymbol\Lambda^\dagger (\mathbf C^\dagger \mathbf S \mathbf C - \mathbf 1)) \Bigr] \;=\; 0 .\]

Differentiating with respect to the MO coefficients $\mathbf C$ gives the Roothaan–Hall equation (chapters 03 and 06), with $\boldsymbol\Lambda$ identified as the diagonal orbital energy matrix $\boldsymbol\varepsilon$. Cross-reference: chapters 03, 06. —

13. The Coulomb kernel $1 / |\mathbf r - \mathbf r’|$ and the Ewald sum

The Coulomb kernel is the single most important special function in DFT. Its Fourier transform, its decomposition into short- and long-range parts, and the Ewald sum that renders the lattice sum of $1/R$ convergent are the three identities every solid-state code needs.

13.1 Fourier transform

The Fourier transform of the Coulomb kernel is

\[\label{eq:ch-extra-fourier-coulomb} \int_{\mathbb R^3} \frac{e^{-i \mathbf k \cdot \mathbf r}}{r}\, d\mathbf r \;=\; \frac{4\pi}{k^2} ,\]

with the convention of §5.2 above (no $1/(2\pi)^{3/2}$ prefactor). Equivalently,

\[\label{eq:ch-extra-poisson} -\nabla^2 \left( \frac{1}{\lvert \mathbf r - \mathbf r' \rvert} \right) \;=\; 4\pi\, \delta(\mathbf r - \mathbf r') ,\]

i.e. $1/r$ is the Green’s function of the Poisson equation in three dimensions. The Fourier-space form $v_\text{Coulomb}(k) = 4\pi / k^2$ is what makes the Hartree operator diagonal in reciprocal space — a $k$-space Hartree potential is just $v_H(\mathbf G) = (4\pi / G^2) \rho(\mathbf G)$, and the plane-wave Hartree-energy evaluation of chapter 07 (§7.5) is therefore an $\mathcal O(N_\text{PW} \log N_\text{PW})$ FFT per SCF iteration. Cross-reference: chapters 04 (§4.4), 07 (§7.5).

13.2 Range separation

The Coulomb kernel can be split into a short-range and a long-range part by an error-function partition

\[\label{eq:ch-extra-range-sep} \frac{1}{r_{12}} \;=\; \underbrace{\frac{\operatorname{erfc}(\omega r_{12})}{r_{12}}}_\text{short range} + \underbrace{\frac{\operatorname{erf}(\omega r_{12})}{r_{12}}}_\text{long range} .\]

The range-separation parameter $\omega$ controls the split: $\omega \to 0$ recovers the full $1/r$ on the long-range side; $\omega \to \infty$ puts everything on the short-range side. The two pieces are treated with different functionals in range-separated hybrid XC functionals (chapter 05, §5.5). Cross-reference: chapter 05. ### 13.3 The Ewald sum

The lattice sum

\[\label{eq:ch-extra-lattice} V_\text{Mad} \;=\; \frac{1}{2} \sum_{\mathbf R \ne 0} \frac{1}{\lvert \mathbf R \rvert}\]

is conditionally convergent (depends on the order of summation). The Ewald decomposition renders it absolutely convergent by writing

\[\label{eq:ch-extra-ewald} \frac{1}{r} \;=\; \underbrace{\frac{\operatorname{erfc}(\alpha r)}{r}}_\text{short range, real space} + \underbrace{\frac{\operatorname{erf}(\alpha r)}{r}}_\text{long range, reciprocal space} .\]

The sum then splits into three pieces:

\(V_\text{Mad} \;=\; \underbrace{\frac{1}{2} \sum_{\mathbf R \ne 0} \frac{\operatorname{erfc}(\alpha \lvert \mathbf R \rvert)}{\lvert \mathbf R \rvert}}_{V_\text{real}} \label{eq:ch-extra-ewald-real}\) \(\quad + \underbrace{\frac{1}{2\Omega} \sum_{\mathbf G \ne 0} \frac{4\pi e^{-\lvert \mathbf G \rvert^2 / 4\alpha^2}}{G^2}}_{V_\text{recip}} \label{eq:ch-extra-ewald-recip}\) \(\quad - \underbrace{\frac{\alpha}{\sqrt\pi}}_{V_\text{self}} . \label{eq:ch-extra-ewald-self}\)

The real-space sum $V_\text{real}$ converges rapidly because $\operatorname{erfc}(\alpha r) \sim e^{-\alpha^2 r^2}$ at large $r$. The reciprocal-space sum $V_\text{recip}$ converges rapidly because $e^{-G^2/4\alpha^2}$ kills large $G$. The self-term $V_\text{self}$ subtracts the spurious interaction of each ion with its own image, introduced when the long-range part is summed in reciprocal space. The parameter $\alpha$ trades real-space and reciprocal-space convergence; the optimal choice is $\alpha \sim \sqrt{\pi}/\text{cell size}$.

The Ewald sum is the Madelung constant of the lattice (for point ions in a compensating background), generalised to finite-size ions and to the electron–ion interaction in periodic DFT. It is the heart of every plane-wave and real-space-grid periodic code. Cross-reference: chapters 04 (the Hartree potential in a periodic cell), 07. ### 13.4 The Madelung constant

The Madelung constant $\mathcal M$ of a Bravais lattice is defined by

\[\label{eq:ch-extra-madelung} V_\text{Mad} \;=\; -\frac{\mathcal M}{2 r_0} ,\]

where $r_0$ is the nearest-neighbour distance. Examples:

Lattice $\mathcal M$
NaCl (rocksalt) 1.74756…
CsCl (bcc) 1.76267…
ZnS (zincblende) 1.63806…

The Ewald sum with the appropriate geometry gives these numbers. Cross-reference: chapter 07 (§7.7, the nearly-free-electron band structure).

14. Common integrals

The chapters need a small set of standard integrals over and over: the Gaussian (basis-set normalisation, the Ewald sum, the path-integral propagator), the Fermi–Dirac and Bose–Einstein distributions (finite-temperature DFT, the electronic entropy of chapter 07), and the polylogarithm that ties them together.

14.1 The Gaussian integral

\[\label{eq:ch-extra-gauss} \int_{-\infty}^{\infty} e^{-a x^2}\, dx \;=\; \sqrt{\frac{\pi}{a}} , \qquad \int_{-\infty}^{\infty} x^{2n}\, e^{-a x^2}\, dx \;=\; \frac{(2n-1)!!}{(2a)^n} \sqrt{\frac{\pi}{a}} .\]

In $d$ dimensions, with $a > 0$,

\[\label{eq:ch-extra-gauss-d} \int_{\mathbb R^d} e^{-a \mathbf x \cdot \mathbf x}\, d^d x \;=\; \left( \frac{\pi}{a} \right)^{d/2} .\]

The Gaussian is the only integral that converges in the free-particle path integral of chapter 01 (§1.11.3, problem 3 of §1.13). It is also the workhorse of the overlap and kinetic-energy integrals of chapter 06 (§6.3, the Gaussian product theorem).

14.2 The error-function integral

\[\label{eq:ch-extra-erfint} \int_0^\infty e^{-a x^2}\, \operatorname{erf}(b x)\, dx \;=\; \frac{1}{2\sqrt{a}}\left[ \frac{\pi}{2} - \arctan\!\left( \frac{\sqrt{a}}{b} \right) \right] \quad (a, b > 0) .\]

This identity is needed when the Coulomb kernel is split by the range-separation of §13.2 above: the short-range contribution to the Hartree energy is a quadrature over Gaussian primitives weighted by an error function. Cross- reference: chapter 05 (§5.5, range-separated hybrids).

14.3 The Fermi–Dirac integral

The Fermi–Dirac occupation of chapter 07 (§7.6.3) is

\[\label{eq:ch-extra-fd} f(\varepsilon) \;=\; \frac{1}{e^{(\varepsilon - \mu)/k_B T} + 1} .\]

The corresponding Fermi–Dirac integral of order $j$ is

\[\label{eq:ch-extra-fdj} \mathcal F_j(\eta) \;=\; \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{x^j}{e^{x - \eta} + 1}\, dx ,\]

with $\eta = \mu / k_B T$ the reduced chemical potential. The Sommerfeld expansion at low temperature gives

\[\label{eq:ch-extra-sommerfeld} \mathcal F_j(\eta) \;=\; \frac{\eta^{j+1}}{(j+1) \Gamma(j+1)} \left[ 1 + \frac{\pi^2}{6}\, j(j+1)\, \eta^{-2} + \mathcal O(\eta^{-4}) \right] .\]

For the density of states $g(\varepsilon)$ of a metal, the electron number is $N = \int_0^\infty g(\varepsilon) f(\varepsilon) d\varepsilon$. The Sommerfeld expansion is the bridge from the low-$T$ behaviour to the $T = 0$ band structure of chapter 07. Cross- reference: chapter 07 (§7.6.3).

14.4 The Bose–Einstein integral

The Bose–Einstein occupation is

\[\label{eq:ch-extra-be} n(\varepsilon) \;=\; \frac{1}{e^{(\varepsilon - \mu)/k_B T} - 1} ,\]

with $\varepsilon > \mu$ (chemical potential below the band edge for a stable Bose gas). The corresponding Bose–Einstein integral of order $j$ is

\[\label{eq:ch-extra-be-j} \mathcal B_j(\eta) \;=\; \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{x^j}{e^{x - \eta} - 1}\, dx \;=\; \operatorname{Li}_{j+1}(e^\eta) \zeta(j+1) / \zeta(j+1) \cdots\]

The Bose–Einstein distribution describes phonons (chapter 10, vibrations in molecules and solids) and the photon gas. The phonon contribution to the free energy of a harmonic crystal is

\[\label{eq:ch-extra-phonon-f} F_\text{ph} \;=\; k_B T \sum_{\mathbf q, s} \ln\!\Bigl(1 - e^{-\hbar\omega_{\mathbf q s}/k_B T}\Bigr) ,\]

where the sum is over phonon branches $s$ and wavevectors $\mathbf q$. Cross-reference: chapter 10 (phonons), chapter 01 (§1.2, P6 for bosons).

14.5 The polylogarithm

The polylogarithm $\operatorname{Li}_s(z)$ is defined by the series

\[\label{eq:ch-extra-polylog} \operatorname{Li}_s(z) \;=\; \sum_{k=1}^{\infty} \frac{z^k}{k^s} \;=\; \frac{z}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}}{e^t / z - 1}\, dt ,\]

convergent for $|z| \le 1$ (real $s > 0$; analytic continuation elsewhere). The integral form is the Bose–Einstein integral of the previous section. Special values: $\operatorname{Li}1(z) = -\ln(1 - z)$, $\operatorname{Li}_2(1) = \pi^2/6$, $\operatorname{Li}_3(1) = \zeta(3) \approx 1.20206$, $\operatorname{Li}_s(1) = \zeta(s)$ (the Riemann zeta function). The Fermi–Dirac integral is $\mathcal F_j(\eta) = -\operatorname{Li}{j+1}(-e^\eta)$. Cross- reference: chapter 07 (finite-temperature occupation numbers).

14.6 The Bose–Einstein condensation temperature

For a three-dimensional non-interacting Bose gas of $N$ particles in a volume $V$ with mass $m$,

\[\label{eq:ch-extra-bec-tc} k_B T_c \;=\; \frac{2\pi \hbar^2}{m} \left( \frac{N}{V\, \zeta(3/2)} \right)^{2/3} ,\]

where $\zeta(3/2) \approx 2.612$ is the Riemann zeta function. This is the textbook result that anchors every discussion of macroscopic quantum phenomena in the notes. The DFT chapters do not derive it, but it appears in the “what we left out” sections as the canonical example of a broken-symmetry ground state beyond the reach of single-Slater-determinant theory. Cross-reference: chapter 01 (§1.14, identical-particle statistics for bosons).

14.7 The Planck (black-body) integral

\[\label{eq:ch-extra-planck} \int_0^\infty \frac{x^3}{e^x - 1}\, dx \;=\; \frac{\pi^4}{15} .\]

This is the Stefan–Boltzmann constant in disguise: $\sigma = (2\pi^5 k_B^4)/(15 h^3 c^2)$. It is the integral that sets the scale of the zero-point radiation that gives the Lamb shift of chapter 01 (§1.14, “what we left out”). Cross- reference: chapter 01. ### 14.8 The complete elliptic integrals

The two complete elliptic integrals of the first and second kind,

\(K(k) \;=\; \int_0^{\pi/2} \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}} , \label{eq:ch-extra-ellip-k}\) \(E(k) \;=\; \int_0^{\pi/2} \sqrt{1 - k^2 \sin^2 \theta}\, d\theta , \label{eq:ch-extra-ellip-e}\)

appear in the 2-D Coulomb problem and in the image-charge sums of surface science. They are not in the main text of the DFT Notes but are the right tool when the geometry of a problem is planar (a graphene sheet, a 2-D electron gas, a slab in a surface calculation). The values at $k = 1$ are $K(1) = \infty$ (logarithmic divergence) and $E(1) = 1$.

Where to look for what you forgot