Equations

Centralized equation catalog. Reference any of these from any page with <Eq id="..."/> or, in the audit textarea, $$@id$$. Edit definitions in src/data/equations.ts.

1D box eigenfunctions

ψn(x)=2L sin⁡ ⁣(nπxL)\psi_n(x) = \sqrt{\frac{2}{L}}\,\sin\!\left(\frac{n\pi x}{L}\right)

Normalized stationary states of a particle confined to (0,L) with infinite walls. The textbook example of energy quantisation from boundary conditions.

particle-in-box-eigenfunction quantum-mechanics

1D harmonic oscillator (atomic units)

−12d2ψdx2+12ω2x2ψ=Eψ-\tfrac{1}{2}\frac{d^2\psi}{dx^2} + \tfrac{1}{2}\omega^2 x^2\psi = E\psi

Time-independent Schrödinger equation for the 1D harmonic oscillator in atomic units. Reference benchmark for finite-difference and spectral solvers.

harmonic-oscillator-1d-tise quantum-mechanics

1s Slater-type orbital

ψSTO(r)=(ζ3π)1/2e−ζr\psi_{\text{STO}}(r) = \left(\frac{\zeta^3}{\pi}\right)^{1/2} e^{-\zeta r}

Hydrogen-like 1s orbital with adjustable exponent ζ. The variational wavefunction whose minimum gives ζ_opt(R) = 1 for the free hydrogen atom.

sto-1s quantum-chemistry

2D Laplace finite-difference stencil

ui,j=14(ui+1,j+ui−1,j+ui,j+1+ui,j−1)u_{i,j} = \tfrac{1}{4}\bigl(u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1}\bigr)

Discrete five-point Laplacian on a uniform grid. The fixed-point form of the discrete Poisson/Laplace problem and the basis for Jacobi and Gauss–Seidel iterations.

finite-difference-laplace-2d numerical-methodspde

3D isotropic harmonic oscillator energies

Enl=ℏω ⁣(2n+l+32)E_{n l} = \hbar\omega\!\left(2 n + l + \tfrac{3}{2}\right)

Spectrum of the 3D isotropic harmonic oscillator labelled by radial and orbital quantum numbers. The starting point of the nuclear shell model.

isotropic-3d-ho nuclear-physicsquantum-mechanics

3D isotropic harmonic oscillator levels

Enl=ℏω(2n+l+32)E_{nl} = \hbar\omega\left(2n + l + \tfrac{3}{2}\right)

Single-particle energy levels of the 3D isotropic oscillator labelled by radial n and orbital l. Foundation of the harmonic-oscillator shell-model basis.

isotropic-3d-ho-shell-model nuclear-physicsquantum-mechanics

7-point 3D Laplacian

∇2f≈f(x+h,y,z)+f(x−h,y,z)+f(x,y+h,z)+f(x,y−h,z)+f(x,y,z+h)+f(x,y,z−h)−6f(x,y,z)h2\nabla^2 f \approx \frac{f(x+h,y,z) + f(x-h,y,z) + f(x,y+h,z) + f(x,y-h,z) + f(x,y,z+h) + f(x,y,z-h) - 6 f(x,y,z)}{h^2}

Standard 7-point stencil for the Laplacian on a 3D Cartesian grid. The 3D analogue of the 5-point 2D star.

finite-difference-laplace-3d numerical-methodspde

Adomian decomposition

L[u]+R[u]+N[u]=f(x)  ⟹  u=L−1 ⁣[−R[u]−N[u]+f(x)]L[u] + R[u] + N[u] = f(x) \;\Longrightarrow\; u = L^{-1}\!\bigl[-R[u] - N[u] + f(x)\bigr]

Splits an operator equation into linear, remainder, and nonlinear parts and solves it as a series in Adomian polynomials. Useful for nonlinear ODEs and PDEs.

adomian-decomposition differential-equationsseries

Adomian polynomial

An=1n!dndλn ⁣[q ⁣(x,∑k=0nvkλk)]λ=0A_n = \frac{1}{n!}\frac{d^n}{d\lambda^n}\!\left[q\!\left(x, \sum_{k=0}^{n} v_k \lambda^k\right)\right]_{\lambda=0}

Generating-function definition of the n-th Adomian polynomial used to expand nonlinear operators in Adomian's decomposition.

adomian-recurrence differential-equationsseries

Antisymmetrized two-electron integral

⟨pq∣∣rs⟩=⟨pq∣rs⟩−⟨pq∣sr⟩\langle pq || rs \rangle = \langle pq | rs \rangle - \langle pq | sr \rangle

Direct minus exchange part of a two-electron integral in physicist notation; the natural object appearing in Slater–Condon rules and FCI matrix elements.

antisymmetrized-two-electron-integral quantum-chemistry

AO overlap matrix element

Sμν=⟨ϕμ∣ϕν⟩=∫ϕμ∗(r) ϕν(r) drS_{\mu\nu} = \langle \phi_\mu | \phi_\nu \rangle = \int \phi_\mu^*(\mathbf r)\,\phi_\nu(\mathbf r)\,d\mathbf r

Inner product of two basis functions. Equal to the identity for orthonormal bases; appears as the right-hand side metric in the Roothaan–Hall generalized eigenvalue problem.

overlap-matrix-element quantum-chemistrylinear-algebra

APW matching coefficients

Clm=Ω−1/2 4πil Ylm∗(k^) jl(krMT)ul(rMT,E)C_{lm} = \Omega^{-1/2}\,4\pi i^l\,Y_l^{m*}(\hat{\mathbf k})\,\frac{j_l(k r_{MT})}{u_l(r_{MT}, E)}

Coefficients that match the plane-wave outside region to the muffin-tin radial part at the sphere boundary. Derived from the Rayleigh expansion.

apw-matching-coefficients solid-state-physics

Augmented plane wave basis function

ϕk(r)={∑l,mClm ul(r,E) Ylm(r^),∣r−Rα∣<SαΩ−1/2eik⋅r,∣r−Rα∣≥Sα\phi_{\mathbf k}(\mathbf r) = \begin{cases} \sum_{l,m} C_{lm}\,u_l(r,E)\,Y_l^m(\hat{\mathbf r}), & |\mathbf r - \mathbf R_\alpha| < S_\alpha \\ \Omega^{-1/2} e^{i\mathbf k\cdot\mathbf r}, & |\mathbf r - \mathbf R_\alpha| \ge S_\alpha \end{cases}

Augmented plane wave: atomic-like radial × spherical-harmonic inside muffin-tin spheres, plain plane wave between them. Backbone of the original APW method.

apw-basis solid-state-physics

Bernoulli differential equation

y′+p(x) y=g(x) yαy' + p(x)\,y = g(x)\,y^{\alpha}

A nonlinear first-order ODE that linearizes under the substitution v = y^{1-α}, giving a closed-form solution in terms of an integrating factor.

bernoulli-equation differential-equations

Bernoulli equation

dydx+p(x)y=g(x)yα\frac{dy}{dx} + p(x) y = g(x) y^{\alpha}

Nonlinear ODE that becomes linear after the substitution v = y^{1−α}. The textbook bridge from linear to nonlinear ODEs.

bernoulli-ode differential-equations

Binder cumulant

UL=1−⟨m4⟩L3 ⟨m2⟩L2U_L = 1 - \frac{\langle m^{4}\rangle_L}{3\,\langle m^{2}\rangle_L^{2}}

Dimensionless ratio of magnetisation moments. Curves for different system sizes cross at the critical temperature, making it the standard finite-size estimator of phase transitions.

binder-cumulant statistical-mechanics

Bisection error bound

∣xn−x∗∣≤b−a2n|x_n - x^{*}| \leq \frac{b - a}{2^{n}}

Worst-case error of the bisection method on a sign-changing interval. Linear convergence — slow but utterly reliable.

bisection-error-bound numerical-methods

Black–Scholes call price

C=S0 N(d1)−Ke−rT N(d2),d1=ln⁡(S0/K)+(r+σ2/2)TσT,d2=d1−σTC = S_0\,N(d_1) - K e^{-rT}\,N(d_2),\qquad d_1 = \frac{\ln(S_0/K) + (r + \sigma^{2}/2)T}{\sigma\sqrt{T}},\quad d_2 = d_1 - \sigma\sqrt{T}

Closed-form price of a European call option under geometric Brownian motion. The companion put follows from put–call parity.

black-scholes-call finance

Black–Scholes d1

d1=ln⁡(S0/K)+(r+σ2/2)TσTd_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt T}

First standardized log-moneyness used in the Black–Scholes formula. Increases with moneyness, rate, and volatility.

black-scholes-d1 finance

Black–Scholes d2

d2=d1−σTd_2 = d_1 - \sigma\sqrt T

Volatility-shifted d1. N(d_2) is the risk-neutral probability that the option finishes in the money.

black-scholes-d2 finance

Black–Scholes PDE

∂V∂t+12σ2S2 ∂2V∂S2+rS ∂V∂S−rV=0\frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2\,\frac{\partial^2 V}{\partial S^2} + r S\,\frac{\partial V}{\partial S} - rV = 0

Partial differential equation governing the price V of a European option under geometric Brownian motion of the underlying.

black-scholes financepde

Black–Scholes put price

P(S,K,T,r,σ)=Ke−rTN(−d2)−S0N(−d1)P(S, K, T, r, \sigma) = K e^{-rT} N(-d_2) - S_0 N(-d_1)

Closed-form European put price under the Black–Scholes assumptions. The put-call parity counterpart of the call formula.

black-scholes-put finance

Black–Scholes vega

ν=∂V∂σ=S0T N′(d1)\nu = \frac{\partial V}{\partial \sigma} = S_0\sqrt{T}\,N'(d_1)

Sensitivity of a European option price to volatility. Drives the Newton step in implied-volatility solvers and is the headline risk hedged by volatility traders.

vega finance

Bloch wave-vector translation

kn=k+Kn\mathbf k_n = \mathbf k + \mathbf K_n

Each plane-wave component in a periodic basis combines the crystal momentum k with a reciprocal-lattice vector K_n. The way Brillouin-zone folding shows up at the basis level.

lapw-bloch-momentum solid-state-physics

Born probability density

ρ(r,t)=∣ψ(r,t)∣2\rho(\mathbf r, t) = |\psi(\mathbf r, t)|^2

Probability density for finding a particle at position r at time t. The Born interpretation of the wavefunction.

wavefunction-probability-density quantum-mechanics

Born probability density

ρ(r,t)=∣ψ(r,t)∣2\rho(\mathbf r, t) = |\psi(\mathbf r, t)|^2

Born's interpretation: the modulus-squared of the wavefunction is the probability density for finding the particle at position r at time t.

born-probability-density quantum-mechanics

Boys F0 function

F0(t)=∫01e−ts2 ds=π4t erf(t)F_0(t) = \int_0^1 e^{-t s^2}\,ds = \sqrt{\frac{\pi}{4 t}}\,\text{erf}(\sqrt t)

Special function appearing in every Gaussian two-electron integral. Reduces to an error function and is the n=0 case of the Boys family F_m(t).

fermi-incomplete-gamma quantum-chemistry

Bravais lattice vectors

Rn=n1a1+n2a2+n3a3\mathbf R_n = n_1\mathbf a_1 + n_2\mathbf a_2 + n_3\mathbf a_3

Integer combinations of three primitive vectors generating the periodic point set of a crystal. The structural foundation of solid-state physics.

bravais-lattice solid-state-physics

Cartesian Gaussian basis function

ϕ(r)=xlymzne−αr2\phi(\mathbf r) = x^l y^m z^n e^{-\alpha r^2}

Workhorse primitive of molecular electronic structure. Polynomial prefactor controls angular momentum; Gaussian width is set by α.

gto-cartesian quantum-chemistry

Central second-difference (psi notation)

d2ψdx2≈ψi+1−2ψi+ψi−1(Δx)2\frac{d^2\psi}{dx^2} \approx \frac{\psi_{i+1} - 2\psi_i + \psi_{i-1}}{(\Delta x)^2}

Three-point central stencil for the second derivative on a uniform grid. The discretization that turns a 1D Schrödinger equation into a tridiagonal eigenvalue problem.

central-difference-second-deriv-grid numerical-methodspde

Central-difference second derivative

f′′(x)≈f(x+h)−2f(x)+f(x−h)h2f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^{2}}

The standard three-point second-derivative stencil. The workhorse of finite-difference discretizations for diffusion, Schrödinger, and Poisson problems.

central-difference-second-derivative numerical-methodsdifferential-equations

Cesàro (Fejér) average

σN(x)=1N+1∑k=0NSk(x)\sigma_N(x) = \frac{1}{N+1}\sum_{k=0}^{N} S_k(x)

Average of the first N partial Fourier sums. Recovers convergence at jump discontinuities where the raw Fourier series fails (Gibbs phenomenon).

cesaro-partial-sum series

Cesàro mean

σN(x)=1N+1∑k=0NSk(x)\sigma_N(x) = \frac{1}{N+1}\sum_{k=0}^{N} S_k(x)

Average of partial sums. Tames Gibbs oscillations and assigns sensible values to series that diverge in the ordinary sense.

cesaro-sum series

Chebyshev expansion

P(x)=∑k=0nckTk(x)P(x) = \sum_{k=0}^{n} c_k T_k(x)

Approximate a function as a finite sum of Chebyshev polynomials. Best near-minimax polynomial of its degree.

chebyshev-poly-expansion series

Chebyshev polynomial recurrence

T0(x)=1,T1(x)=x,Tn+1(x)=2x Tn(x)−Tn−1(x)T_0(x) = 1,\quad T_1(x) = x,\quad T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x)

Three-term recurrence for the Chebyshev polynomials of the first kind. Underlies Chebyshev interpolation, near-minimax polynomial approximation, and spectral collocation.

chebyshev-recurrence series

Chebyshev polynomials of the first kind

T0(x)=1,T1(x)=x,Tn+1(x)=2x Tn(x)−Tn−1(x)T_0(x) = 1, \quad T_1(x) = x, \quad T_{n+1}(x) = 2 x\,T_n(x) - T_{n-1}(x)

Standard three-term recurrence generating the Chebyshev polynomials. Fundamental to Chebyshev interpolation and spectral methods on [-1, 1].

chebyshev-polynomials-tn series

Chebyshev–Gauss–Lobatto nodes

xk=cos⁡ ⁣(πkN−1),k=0,1,…,N−1x_k = \cos\!\left(\tfrac{\pi k}{N-1}\right),\qquad k = 0, 1, \ldots, N-1

Cosine-spaced collocation grid that clusters near the endpoints and avoids the Runge phenomenon. The natural mesh for Chebyshev spectral methods.

chebyshev-extrema-grid numerical-methods

CI Hamiltonian matrix element (second-quantised)

HIJ=∑pqhpq ⟨ΦI∣ap†aq∣ΦJ⟩+12∑pqrs⟨pq∣∣rs⟩ ⟨ΦI∣ap†aq†asar∣ΦJ⟩H_{IJ} = \sum_{pq} h_{pq}\,\langle\Phi_I| a_p^\dagger a_q |\Phi_J\rangle + \tfrac{1}{2}\sum_{pqrs} \langle pq||rs\rangle\,\langle\Phi_I| a_p^\dagger a_q^\dagger a_s a_r |\Phi_J\rangle

Configuration-interaction matrix element written in second quantisation. Gets reduced to one- and two-particle integrals via Slater–Condon / Wick contractions.

fci-hamiltonian-matrix-element quantum-chemistry

Classical Runge–Kutta (RK4)

k1=f(tn,yn)k2=f ⁣(tn+h2, yn+h2k1)k3=f ⁣(tn+h2, yn+h2k2)k4=f(tn+h, yn+h k3)yn+1=yn+h6(k1+2k2+2k3+k4)\begin{aligned} k_1 &= f(t_n, y_n) \\ k_2 &= f\!\left(t_n + \tfrac{h}{2},\, y_n + \tfrac{h}{2}k_1\right) \\ k_3 &= f\!\left(t_n + \tfrac{h}{2},\, y_n + \tfrac{h}{2}k_2\right) \\ k_4 &= f(t_n + h,\, y_n + h\,k_3) \\ y_{n+1} &= y_n + \tfrac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4) \end{aligned}

The fourth-order Runge–Kutta scheme. Hits a sweet spot between accuracy and simplicity and is the default explicit ODE integrator in countless codes.

runge-kutta-rk4 numerical-methodsdifferential-equations

Closed-shell density matrix

Pμν=2∑i=1N/2Cμi Cνi∗P_{\mu\nu} = 2\sum_{i=1}^{N/2} C_{\mu i}\,C_{\nu i}^{*}

AO-basis one-electron density matrix for a restricted closed-shell wavefunction. Encodes the doubly occupied orbitals and is what the Fock build consumes each SCF iteration.

density-matrix-rhf quantum-chemistry

Closed-shell density matrix

Pμν=2∑i=1N/2CμiCνi∗P_{\mu\nu} = 2\sum_{i=1}^{N/2} C_{\mu i} C_{\nu i}^*

Restricted Hartree–Fock one-particle density matrix in the AO basis. The factor of 2 accounts for double occupancy of each spatial orbital.

rhf-density-matrix quantum-chemistry

Coefficient of determination R^2

R2=(∑xiyi−∑xi∑yi/n)2(∑xi2−(∑xi)2/n)(∑yi2−(∑yi)2/n)R^2 = \frac{(\sum x_i y_i - \sum x_i \sum y_i / n)^2}{(\sum x_i^2 - (\sum x_i)^2 / n)(\sum y_i^2 - (\sum y_i)^2 / n)}

Squared Pearson correlation written in the form a regression code computes it. Tells you what fraction of variance the linear fit explains.

r-squared linear-algebraoptimization

Composite trapezoidal rule

I≈h2[f(a)+2∑i=1N−1f(a+ih)+f(b)]I \approx \frac{h}{2}\left[f(a) + 2\sum_{i=1}^{N-1} f(a + i h) + f(b)\right]

Uniform-grid extension of the basic trapezoidal rule. Endpoints get half weight; second-order accurate in the spacing h.

composite-trapezoidal numerical-methods

Core Hamiltonian matrix element

Hμνcore=⟨ϕμ∣h^∣ϕν⟩H^{\text{core}}_{\mu\nu} = \langle \phi_\mu | \hat h | \phi_\nu \rangle

AO-basis matrix of the one-electron part of the Hamiltonian. Computed once at the start of an SCF run and reused every iteration.

core-hamiltonian-matrix-element quantum-chemistry

Correlation energy

Ecorr=EFCI−EHFE_{\text{corr}} = E_{\text{FCI}} - E_{\text{HF}}

Difference between the exact (FCI) and Hartree–Fock energies in a given basis. The quantity post-HF methods aim to recover.

fci-correlation-energy quantum-chemistry

Correlation-length critical scaling

ξ∼∣T−Tc∣−ν\xi \sim |T - T_c|^{-\nu}

Correlation length diverges at the critical point with exponent ν. ν = 1 for the 2D Ising model.

critical-correlation-length statistical-mechanics

Coulomb operator

J^j(1) ψi(1)=[∫ψj∗(2) 1r12 ψj(2) dr2]ψi(1)\hat J_j(1)\,\psi_i(1) = \left[\int \psi_j^*(2)\,\frac{1}{r_{12}}\,\psi_j(2)\,d\mathbf r_2\right]\psi_i(1)

Local mean-field operator capturing the classical Coulomb repulsion an electron feels from the charge density of orbital j. Direct part of the Hartree–Fock potential.

coulomb-operator quantum-chemistry

Covariance matrix

Σij=Cov⁡(Xi,Xj)=1n∑k=1n(xki−xˉi)(xkj−xˉj)\Sigma_{ij} = \operatorname{Cov}(X_i, X_j) = \tfrac{1}{n}\sum_{k=1}^{n}(x_{ki} - \bar x_i)(x_{kj} - \bar x_j)

Symmetric positive-semidefinite matrix of pairwise covariances. The starting point for PCA, Mahalanobis distance, and Gaussian likelihoods.

covariance-matrix linear-algebra

Covariance matrix entry

Σij=Cov(Xi,Xj)\Sigma_{ij} = \text{Cov}(X_i, X_j)

Symmetric positive-semidefinite matrix collecting all pairwise covariances of a random vector.

covariance-matrix-entry linear-algebra

Davidson correction equation

t=(μI−D)−1 r\mathbf t = (\mu I - D)^{-1}\,\mathbf r

Diagonal-preconditioned correction vector that expands the Davidson subspace. Lets large, sparse, diagonally dominant eigenproblems converge an order of magnitude faster than Lanczos in quantum chemistry.

davidson-correction linear-algebraeigenvaluequantum-chemistry

Davidson projected eigenproblem

Hk=VkTAVk,Hky=μyH_k = V_k^T A V_k, \quad H_k\mathbf y = \mu\mathbf y

Small Rayleigh–Ritz problem solved at each Davidson iteration on the current subspace V_k.

davidson-projected linear-algebraeigenvalue

Davidson residual

r=(μI−A)x\mathbf r = (\mu I - A)\mathbf x

Residual of a Ritz pair (μ, x) used both as a stopping criterion and as the input to Davidson's diagonal preconditioner.

davidson-residual linear-algebraeigenvalue

Eigenvalue equation

A vi=λi viA\,\mathbf v_i = \lambda_i\,\mathbf v_i

Defines the spectrum of a linear operator. The fundamental object for stability, normal modes, principal components, and quantum observables.

eigenvalue-equation linear-algebraeigenvalue

Empirical nuclear radius

R=r0A1/3R = r_0 A^{1/3}

Rule of thumb that nuclei have approximately constant density: radius scales as the cube root of mass number A, with r_0 ≈ 1.2 fm.

nuclear-radius nuclear-physics

Euclidean (l2) norm

∥x∥=x12+x22+⋯+xm2\|x\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_m^2}

Standard length of a real vector. Induced by the dot product and preserved under orthogonal transformations.

euclidean-norm linear-algebra

Euler–Maruyama scheme

yn+1=yn+μ(yn,tn) Δt+σ(yn,tn) ΔWny_{n+1} = y_n + \mu(y_n,t_n)\,\Delta t + \sigma(y_n,t_n)\,\Delta W_n

Lowest-order strong scheme for general SDEs: explicit Euler in the drift plus a Brownian increment scaled by the diffusion. Strong order 0.5.

euler-maruyama financedifferential-equationsalgorithms

European call payoff

Payoff=max⁡(ST−K,0)\text{Payoff} = \max(S_T - K, 0)

Terminal payoff of a vanilla long call: positive only when the underlying finishes above the strike.

option-payoff-call finance

European put payoff

Payoff=max⁡(K−ST,0)\text{Payoff} = \max(K - S_T, 0)

Terminal payoff of a vanilla long put: positive only when the underlying finishes below the strike.

option-payoff-put finance

Exchange operator

K^j(1) ψi(1)=[∫ψj∗(2) 1r12 ψi(2) dr2]ψj(1)\hat K_j(1)\,\psi_i(1) = \left[\int \psi_j^*(2)\,\frac{1}{r_{12}}\,\psi_i(2)\,d\mathbf r_2\right]\psi_j(1)

Non-local one-body operator with no classical analogue; a direct consequence of antisymmetrizing fermionic wavefunctions and responsible for Pauli (Fermi) correlation between same-spin electrons.

exchange-operator quantum-chemistry

Explicit (forward) Euler method

yn+1=yn+h f(tn,yn)y_{n+1} = y_n + h\,f(t_n, y_n)

Simplest first-order time-stepping scheme for an initial-value ODE. Conditionally stable and the entry point to every more sophisticated integrator.

explicit-euler numerical-methodsdifferential-equations

FCI configuration count

Nconfig=(MNα)(MNβ)N_{\text{config}} = \binom{M}{N_\alpha}\binom{M}{N_\beta}

Number of Slater determinants in a full-CI expansion with M spatial orbitals and given α/β electron counts. Grows factorially and is the reason FCI is feasible only for tiny systems.

fci-dimension quantum-chemistry

FFT spectral derivative

dfdx∣j=Re ⁣[IFFT ⁣(ikj FFT(fj))]\frac{df}{dx}\bigg|_j = \text{Re}\!\left[\text{IFFT}\!\left(i k_j\,\text{FFT}(f_j)\right)\right]

Compute the derivative of a periodic function in O(N log N) by multiplying its Fourier coefficients by ik. Spectrally accurate for smooth f.

fft-spectral-derivative seriesnumerical-methodsalgorithms

FFT wavenumber grid

kj=2πjN Δx,j=−N2,…,N2−1k_j = \frac{2\pi j}{N\,\Delta x}, \quad j = -\tfrac{N}{2},\ldots,\tfrac{N}{2}-1

Conjugate Fourier wavenumbers for an N-point uniform grid of spacing Δx. The frequencies the FFT really sees.

spectral-wavenumber-grid seriesnumerical-methods

Fibonacci recurrence

Fn=Fn−1+Fn−2for n≥2,F0=0,  F1=1F_n = F_{n-1} + F_{n-2}\quad\text{for } n \ge 2,\quad F_0 = 0,\;F_1 = 1

Defining recurrence of the Fibonacci sequence. The textbook example for memoised dynamic programming.

fibonacci-recurrence seriesalgorithms

Fixed-point iteration

x=f(x)x = f(x)

Generic recasting of an equation into iteration-friendly form. Converges whenever f is a contraction on a closed neighbourhood of the fixed point.

fixed-point-iteration numerical-methodsalgorithms

Fock eigenvalue equation

F^ ∣ψi⟩=ϵi ∣ψi⟩\hat F\,|\psi_i\rangle = \epsilon_i\,|\psi_i\rangle

Canonical eigenvalue form of the Hartree–Fock equations: orbital energies as eigenvalues of the Fock operator on its own eigenfunctions.

fock-eigenvalue-equation quantum-chemistryeigenvalue

Fock matrix element (AO basis)

Fμν=Hμνcore+∑λσPλσ[(μν∣λσ)−12(μλ∣νσ)]F_{\mu\nu} = H^{\text{core}}_{\mu\nu} + \sum_{\lambda\sigma} P_{\lambda\sigma}\left[(\mu\nu|\lambda\sigma) - \tfrac{1}{2}(\mu\lambda|\nu\sigma)\right]

AO-basis Fock matrix expressed as core Hamiltonian plus a density-weighted contraction of two-electron integrals. The object that gets diagonalized every SCF step.

fock-matrix-element-ao quantum-chemistry

Fock matrix in the AO basis

Fμν=Hμνcore+∑λσPλσ ⁣[(μν∣λσ)−12(μλ∣νσ)]F_{\mu\nu} = H_{\mu\nu}^{\text{core}} + \sum_{\lambda\sigma} P_{\lambda\sigma}\!\left[(\mu\nu|\lambda\sigma) - \tfrac{1}{2}(\mu\lambda|\nu\sigma)\right]

Closed-shell Fock matrix written from the density matrix and two-electron integrals. The object that gets diagonalized at every SCF cycle.

fock-matrix-elements quantum-chemistry

Fock operator

F^(1)=h^(1)+∑j=1N/2 [2J^j(1)−K^j(1)]\hat F(1) = \hat h(1) + \sum_{j=1}^{N/2}\,[2\hat J_j(1) - \hat K_j(1)]

One-electron operator that defines the mean-field Hartree–Fock problem: core Hamiltonian plus a sum of Coulomb and exchange contributions from every other occupied orbital.

fock-operator quantum-chemistryhartree-fock

Forward Euler step

yn+1=yn+h f(xn,yn)y_{n+1} = y_n + h\,f(x_n, y_n)

Simplest explicit ODE integrator: take a step of size h along the slope at the current point. First-order accurate, conditionally stable.

forward-euler differential-equationsnumerical-methods

Fourier coefficient a0

a0=2P∫0Pf(x) dxa_0 = \frac{2}{P}\int_0^P f(x)\,dx

The DC coefficient of the real Fourier series — twice the average value of f over a period.

fourier-coeff-a0 series

Fourier coefficient formulas

an=2P∫0Pf(x)cos⁡ ⁣(2nπxP)dx,bn=2P∫0Pf(x)sin⁡ ⁣(2nπxP)dxa_n = \frac{2}{P}\int_0^{P} f(x)\cos\!\left(\tfrac{2 n \pi x}{P}\right)dx,\qquad b_n = \frac{2}{P}\int_0^{P} f(x)\sin\!\left(\tfrac{2 n \pi x}{P}\right)dx

Inner-product projections recovering the cosine and sine amplitudes of a periodic function. Orthogonality of the trigonometric basis makes them independent.

fourier-coefficients series

Fourier cosine coefficients

an=2P∫0Pf(x)cos⁡ ⁣(2nπxP)dxa_n = \frac{2}{P}\int_0^P f(x)\cos\!\left(\frac{2 n \pi x}{P}\right) dx

Projection of f onto the n-th cosine mode of period P, normalized so the series formula carries no extra factor.

fourier-coeff-an series

Fourier series

f(x)∼a02+∑n=1∞ ⁣[ancos⁡ ⁣(2nπxP)+bnsin⁡ ⁣(2nπxP)]f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty}\!\left[a_n \cos\!\left(\tfrac{2 n \pi x}{P}\right) + b_n \sin\!\left(\tfrac{2 n \pi x}{P}\right)\right]

Decomposes a periodic function into orthogonal sinusoids. The mother of every spectral method and the language of signal processing.

fourier-series series

Fourier sine coefficients

bn=2P∫0Pf(x)sin⁡ ⁣(2nπxP)dxb_n = \frac{2}{P}\int_0^P f(x)\sin\!\left(\frac{2 n \pi x}{P}\right) dx

Projection of f onto the n-th sine mode of period P. Vanishes for even f, so symmetry kills half the work.

fourier-coeff-bn series

Fourier spectral derivative

dfdx=F−1 ⁣[ik F[f]]\frac{df}{dx} = \mathcal F^{-1}\!\bigl[i k\, \mathcal F[f]\bigr]

Exponentially convergent differentiation for smooth periodic functions: multiply by ik in Fourier space. The basis for pseudospectral PDE solvers.

fourier-spectral-derivative series

Free-particle TISE in wavenumber form

d2dx2ψ(x)=−2mEℏ2ψ(x)\frac{d^2}{dx^2}\psi(x) = -\frac{2mE}{\hbar^2}\psi(x)

Schrödinger equation for V = 0, written so that k^2 = 2mE/hbar^2 makes the dispersion relation explicit.

free-particle-wavenumber quantum-mechanics

Full configuration interaction expansion

∣ΨFCI⟩=∑IcI ∣ΦI⟩|\Psi_{\text{FCI}}\rangle = \sum_I c_I\,|\Phi_I\rangle

Exact wavefunction in a finite basis: a linear combination of every Slater determinant compatible with the spin and particle counts. The benchmark all approximate methods aim to recover.

fci-expansion quantum-chemistry

Gauss–Seidel update for Laplace

ui,j(k+1)=14(ui+1,j(k)+ui−1,j(k+1)+ui,j+1(k)+ui,j−1(k+1))u_{i,j}^{(k+1)} = \tfrac{1}{4}\bigl(u_{i+1,j}^{(k)} + u_{i-1,j}^{(k+1)} + u_{i,j+1}^{(k)} + u_{i,j-1}^{(k+1)}\bigr)

Gauss–Seidel sweep for the discrete 2D Laplace equation. Uses already-updated neighbors in place, roughly doubling convergence speed over Jacobi.

gauss-seidel-laplace numerical-methodspde

Gaussian product centre

Rp=α RA+β RBα+β\mathbf R_p = \frac{\alpha\,\mathbf R_A + \beta\,\mathbf R_B}{\alpha + \beta}

Centre of the Gaussian formed by multiplying two Gaussians sitting at R_A and R_B. Combined with the product-theorem prefactor it makes Gaussian integrals tractable.

gaussian-product-center quantum-chemistry

Gaussian product theorem

gα(r−RA) gβ(r−RB)=K gα+β(r−Rp),K=exp⁡ ⁣[−αβα+β∣RA−RB∣2]g_{\alpha}(\mathbf r - \mathbf R_A)\,g_{\beta}(\mathbf r - \mathbf R_B) = K\,g_{\alpha+\beta}(\mathbf r - \mathbf R_p),\quad K = \exp\!\left[-\tfrac{\alpha\beta}{\alpha+\beta}|\mathbf R_A - \mathbf R_B|^{2}\right]

A product of two Gaussians is a third Gaussian centred at the weighted midpoint, scaled by an exponential prefactor. The reason quantum chemistry abandoned Slater functions for Gaussian basis sets.

gaussian-product-theorem quantum-chemistry

GBM exact discrete update

St+Δt=Stexp⁡ ⁣( ⁣(r−σ22)Δt+σΔt Z)S_{t+\Delta t} = S_t\exp\!\left(\!\left(r - \tfrac{\sigma^2}{2}\right)\Delta t + \sigma\sqrt{\Delta t}\,Z\right)

Risk-neutral GBM step using a standard normal Z. Distribution-exact at any Δt, so no discretization error in Monte Carlo option pricing.

gbm-discrete-update financealgorithms

Generalized eigenvalue problem

Av=λBvA\mathbf v = \lambda B\mathbf v

Eigenvalue problem with a non-identity right-hand side metric. Arises whenever basis vectors are non-orthogonal (e.g. AO basis in HF).

generalized-eigenvalue linear-algebraeigenvalue

Geometric Brownian motion SDE

dSt=μ St dt+σ St dWtdS_t = \mu\,S_t\,dt + \sigma\,S_t\,dW_t

Stochastic differential equation for log-normal asset prices: drift μ plus volatility σ driven by a Wiener process. The reference dynamics for Black–Scholes and Monte Carlo option pricing.

gbm-sde financedifferential-equations

Geometric Brownian motion solution

St=S0exp⁡ ⁣( ⁣(μ−σ22)t+σWt)S_t = S_0\exp\!\left(\!\left(\mu - \tfrac{\sigma^2}{2}\right)t + \sigma W_t\right)

Closed-form solution of the GBM SDE obtained from Itô's lemma. The −σ²/2 drift correction is the hallmark of stochastic calculus.

gbm-closed-form financedifferential-equations

Geometric series

Sn=a 1−rn1−rS_n = a\,\frac{1 - r^{n}}{1 - r}

Closed form for the sum of a finite geometric progression. Converges to a/(1-r) for |r|<1 and is the analytic backbone of compound interest, perturbation series, and resolvent expansions.

geometric-series series

Geometric series partial sum

Sn=a 1−rn1−rS_n = a\,\frac{1 - r^n}{1 - r}

Closed form for the first n terms of a + ar + ar^2 + .... Letting n→∞ for |r|<1 gives the familiar a/(1−r).

geometric-series-partial-sum series

H2 electronic Hamiltonian (clamped nuclei)

H^=∑i=12h^(i)+1r12+1R\hat H = \sum_{i=1}^{2} \hat h(i) + \frac{1}{r_{12}} + \frac{1}{R}

Two-electron Born–Oppenheimer Hamiltonian for the hydrogen molecule: two one-electron operators, the e–e repulsion, and the (constant) nuclear repulsion.

electronic-hamiltonian-h2 quantum-chemistry

H2+ bonding LCAO orbital

ψg(r)=Ng[ϕ1(r−Ra)+ϕ1(r−Rb)]\psi_g(\mathbf r) = N_g\big[\phi_1(\mathbf r - \mathbf R_a) + \phi_1(\mathbf r - \mathbf R_b)\big]

Symmetric combination of two atomic 1s orbitals — the simplest molecular orbital and the textbook illustration of LCAO and chemical bonding.

h2-mo-bonding quantum-chemistry

Harmonic oscillator energies

En=ℏω ⁣(n+12),n=0,1,2,…E_n = \hbar\omega\!\left(n + \tfrac{1}{2}\right),\qquad n = 0, 1, 2, \ldots

Equally spaced eigenvalues of the quantum harmonic oscillator, including the irreducible zero-point energy. Recovers black-body spectra, vibrational modes, and field quantisation.

harmonic-oscillator-energies quantum-mechanics

Hartree–Fock energy (orbital-energy form)

EHF=∑i=1N/2(ϵi+hii)+VnucE_{\text{HF}} = \sum_{i=1}^{N/2} (\epsilon_i + h_{ii}) + V_{\text{nuc}}

Compact expression of the closed-shell HF energy in terms of orbital eigenvalues and one-electron orbital integrals; useful sanity check after diagonalization.

hartree-fock-energy-orbital-sum quantum-chemistry

Hartree–Fock energy from density matrix

EHF=∑μνPμνHμνcore+12∑μνλσPμνPλσ[(μν∣λσ)−12(μλ∣νσ)]+VnucE_{\text{HF}} = \sum_{\mu\nu} P_{\mu\nu} H^{\text{core}}_{\mu\nu} + \tfrac{1}{2}\sum_{\mu\nu\lambda\sigma} P_{\mu\nu} P_{\lambda\sigma} \left[(\mu\nu|\lambda\sigma) - \tfrac{1}{2}(\mu\lambda|\nu\sigma)\right] + V_{\text{nuc}}

Closed-shell HF total energy expressed entirely in terms of the AO density matrix and pre-computed integrals. The form actually evaluated each SCF iteration.

hartree-fock-energy-density-matrix quantum-chemistry

Householder applied to vector

Hx=−sign⁡(x1)∥x∥ e1H x = -\operatorname{sign}(x_1)\|x\|\,e_1

Defining property of the Householder reflector: it sends x to ±|x|e_1 in one shot, zeroing all entries below the first.

householder-vector-action linear-algebra

Householder rank-1 matrix update

A←A−τ v (vTA)A \leftarrow A - \tau\,v\,(v^T A)

How a Householder reflector is applied without ever forming H explicitly: a scaled outer product subtraction. Cost O(mn) instead of O(m^2 n).

householder-update-matrix linear-algebraalgorithms

Householder reflection vector

v=x+sign⁡(x1)∥x∥ e1v = x + \operatorname{sign}(x_1)\|x\|\,e_1

Choice of v whose reflection sends x onto the first coordinate axis. The sign trick avoids cancellation when x already points along e_1.

householder-vector linear-algebraalgorithms

Householder reflector

H=I−τ vvT,τ=2vTvH = I - \tau\,v v^{T},\qquad \tau = \frac{2}{v^{T}v}

Orthogonal reflection that zeros all but one component of a target vector. The numerically stable building block of QR factorization and tridiagonalisation.

householder-reflector linear-algebra

Hubbard dimer ground-state energy

E0=U2−(U2)2+4t2E_0 = \tfrac{U}{2} - \sqrt{\left(\tfrac{U}{2}\right)^2 + 4t^2}

Closed-form lowest eigenvalue of the singlet block of the two-site Hubbard model. Smoothly interpolates the metallic (U≪t) and Mott (U≫t) limits.

hubbard-dimer-ground-energy quantum-chemistrysolid-state-physics

Hubbard dimer Hamiltonian

H^=−t∑σ(c^1σ†c^2σ+c^2σ†c^1σ)+U∑i=12n^i↑n^i↓\hat H = -t\sum_{\sigma}\bigl(\hat c_{1\sigma}^{\dagger}\hat c_{2\sigma} + \hat c_{2\sigma}^{\dagger}\hat c_{1\sigma}\bigr) + U\sum_{i=1}^{2}\hat n_{i\uparrow}\hat n_{i\downarrow}

Two-site, two-orbital Hubbard model. The minimal lattice toy that captures the competition between hopping (t) and on-site repulsion (U), and a standard test case for correlated-electron methods.

hubbard-dimer solid-state-physicsquantum-chemistry

Hubbard number operator

n^iσ=c^iσ†c^iσ\hat n_{i\sigma} = \hat c_{i\sigma}^\dagger \hat c_{i\sigma}

Counts the number of spin-σ electrons on site i. Building block of every interaction term in lattice fermion models.

hubbard-number-operator quantum-chemistrysolid-state-physics

Implicit (backward) Euler method

yn+1=yn+h f(tn+1,yn+1)y_{n+1} = y_n + h\,f(t_{n+1}, y_{n+1})

First-order implicit scheme that is unconditionally stable for stiff problems, at the cost of solving a (possibly nonlinear) equation at every step.

implicit-euler numerical-methodsdifferential-equations

Implied volatility equation

Vmarket=VBS(S,K,T,r,σimplied)V_{\text{market}} = V_{\text{BS}}(S, K, T, r, \sigma_{\text{implied}})

Definition of implied volatility: the σ that makes Black–Scholes reproduce the market price. Solved with Newton's method or bisection.

implied-volatility-equation financenumerical-methods

Ising model Hamiltonian

H=−J∑⟨i,j⟩sisj−h∑isiH = -J\sum_{\langle i,j\rangle} s_i s_j - h\sum_i s_i

Lattice spin model with nearest-neighbor exchange J and external field h. The simplest system showing a finite-temperature ferromagnetic phase transition and a workhorse of statistical mechanics.

ising-model statistical-mechanics

Ising single-spin flip energy

ΔE=2si(J∑j∈nnsj+h)\Delta E = 2 s_i\left(J\sum_{j\in\text{nn}} s_j + h\right)

Energy change when spin s_i is flipped, given its nearest-neighbor sum and external field h. The quantity tested by the Metropolis acceptance rule.

ising-energy-change statistical-mechanics

Itô stochastic differential equation

dy=μ(y,t) dt+σ(y,t) dWdy = \mu(y,t)\,dt + \sigma(y,t)\,dW

Generic Itô SDE: drift μ plus diffusion σ driving a Wiener increment. Common parent of GBM, Ornstein–Uhlenbeck, CIR, etc.

sde-general financedifferential-equations

Jacobi update for 2D Laplace

ui,j=14(ui+1,j+ui−1,j+ui,j+1+ui,j−1)u_{i,j} = \tfrac{1}{4}\big(u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1}\big)

Discrete mean-value property: at each interior node, the Laplace solution equals the average of its four neighbors. Yields the simplest iterative solver.

jacobi-laplace-update numerical-methodspde

Kahan compensated summation

y=x−ct=s+yc=(t−s)−ys=t\begin{aligned} y &= x - c \\ t &= s + y \\ c &= (t - s) - y \\ s &= t \end{aligned}

Tracks the running rounding error in a compensation variable to recover bits lost when adding numbers of vastly different magnitude.

kahan-summation numerical-methods

Kahan compensated summation

y=x−ct=sum+yc=(t−sum)−ysum=t\begin{aligned} y &= x - c \\ t &= \text{sum} + y \\ c &= (t - \text{sum}) - y \\ \text{sum} &= t \end{aligned}

Inner loop of compensated summation: the running correction c rescues the low-order bits that ordinary floating-point addition would discard.

kahan-loop numerical-methodsalgorithms

Koopmans IP/EA

−ϵi≈IPi-\epsilon_i \approx \text{IP}_i

Negative of an occupied orbital energy approximates the corresponding ionisation potential. Frozen-orbital, no-relaxation estimate from a single HF calculation.

koopmans-affinity quantum-chemistry

Koopmans’ theorem

− ϵi≈IPi-\,\epsilon_i \approx \mathrm{IP}_i

The negative of an occupied orbital energy approximates the ionization potential, neglecting orbital relaxation.

koopmans quantum-chemistry

l1 norm

∥w∥1=∑j∣wj∣\|\mathbf w\|_1 = \sum_j |w_j|

Sum of absolute values. Penalising it (Lasso) drives coefficients to exactly zero, producing sparse models.

l1-norm linear-algebraoptimization

LAPW basis function

ϕkn(r)={∑l,m[alm ul(El,r)+blm u˙l(El,r)]Ylm(r^),∣r−Rα∣<SαΩ−1/2eikn⋅r,∣r−Rα∣≥Sα\phi_{\mathbf k_n}(\mathbf r) = \begin{cases} \sum_{l,m}\big[a_{lm}\,u_l(E_l,r) + b_{lm}\,\dot u_l(E_l,r)\big] Y_l^m(\hat{\mathbf r}), & |\mathbf r - \mathbf R_\alpha| < S_\alpha \\ \Omega^{-1/2} e^{i\mathbf k_n\cdot\mathbf r}, & |\mathbf r - \mathbf R_\alpha| \ge S_\alpha \end{cases}

Linearized augmented plane wave: adds the energy derivative of the radial solution so the basis no longer depends on a fixed linearization energy E_l.

lapw-basis solid-state-physics

LAPW energy-derivative equation

hl u˙l−El u˙l=ulh_l\,\dot u_l - E_l\,\dot u_l = u_l

Inhomogeneous equation defining the energy derivative used as the second LAPW basis component. Solved alongside the radial Schrödinger equation in each muffin-tin.

lapw-energy-derivative solid-state-physics

LAPW radial-derivative coefficient

al=jl′(kR) u˙l(R)−jl(kR) u˙l′(R)a_l = j_l'(kR)\,\dot u_l(R) - j_l(kR)\,\dot u_l'(R)

First LAPW matching coefficient — a 2×2 Wronskian-style determinant ensuring continuity of value and derivative at the muffin-tin radius.

lapw-coefficient-a solid-state-physics

LAPW value coefficient

bl=jl(kR) ul′(R)−jl′(kR) ul(R)b_l = j_l(kR)\,u_l'(R) - j_l'(kR)\,u_l(R)

Second LAPW matching coefficient. Together with a_l it sets the basis function and its slope at the muffin-tin boundary.

lapw-coefficient-b solid-state-physics

Lasso objective

min⁡w,b  1m∑i=1m(yi−wTxi−b)2+λ∥w∥1\min_{\mathbf w, b}\;\frac{1}{m}\sum_{i=1}^{m}\big(y_i - \mathbf w^T \mathbf x_i - b\big)^2 + \lambda\|\mathbf w\|_1

Mean squared error plus an l1 penalty on the weights. The l1 term is what makes Lasso produce exactly-zero coefficients.

lasso-objective optimizationlinear-algebra

Lasso regression

min⁡w,b  1m∑i=1m(yi−wTxi−b)2+λ ∥w∥1\min_{\mathbf w, b}\;\frac{1}{m}\sum_{i=1}^{m}\bigl(y_i - \mathbf w^{T}\mathbf x_i - b\bigr)^{2} + \lambda\,\lVert\mathbf w\rVert_{1}

L1-regularized least squares, due to Tibshirani. The penalty drives many coefficients exactly to zero, performing simultaneous estimation and feature selection.

lasso-regression linear-algebraoptimization

LCAO bonding orbital normalization

Ng=12(1+S)N_g = \frac{1}{\sqrt{2(1+S)}}

Normalization constant for the symmetric two-centre LCAO orbital, S being the overlap integral between the two atomic orbitals.

h2-mo-normalization quantum-chemistry

LCAO molecular orbital expansion

ψi(r)=∑μCμi ϕμ(r)\psi_i(\mathbf r) = \sum_{\mu} C_{\mu i}\,\phi_{\mu}(\mathbf r)

Linear combination of atomic orbitals: expand each MO in a finite atomic-orbital basis. Reduces the differential Hartree–Fock problem to a matrix eigenproblem.

lcao-expansion quantum-chemistry

Legendre polynomial recurrence

Pn(x)=(2n−1)xPn−1(x)−(n−1)Pn−2(x)nP_n(x) = \frac{(2n - 1) x P_{n-1}(x) - (n-1) P_{n-2}(x)}{n}

Bonnet's recurrence, the stable way to compute Legendre polynomials of any degree starting from P_0 = 1, P_1 = x.

legendre-polynomials-pn series

Legendre polynomial recurrence (Bonnet)

Pn(x)=(2n−1) x Pn−1(x)−(n−1) Pn−2(x)nP_n(x) = \frac{(2n-1)\,x\,P_{n-1}(x) - (n-1)\,P_{n-2}(x)}{n}

Bonnet's three-term recurrence for the Legendre polynomials. The numerically stable way to evaluate them and the basis for Gauss–Legendre quadrature.

legendre-recurrence series

Magnetisation critical scaling

M∼∣T−Tc∣βM \sim |T - T_c|^{\beta}

Order-parameter power law near a continuous phase transition. β = 1/8 for the 2D Ising universality class.

critical-magnetisation statistical-mechanics

Matrix eigenvalue equation

Avi=λiviA\mathbf v_i = \lambda_i \mathbf v_i

Definition of an eigenpair of a square matrix. Generalizes to the operator equation that anchors much of physics and numerical linear algebra.

eigenvalue-equation-matrix linear-algebraeigenvalue

Matrix transpose (entrywise)

(AT)ij=Aji(A^T)_{ij} = A_{ji}

Definition of the transpose: row i of A becomes column i of A^T. The most basic shape-preserving symmetry of matrices.

transpose-entrywise linear-algebra

Matrix-vector product (index form)

yi=∑jaij xjy_i = \sum_j a_{ij}\,x_j

Index-by-index definition of y = A x. The atomic operation behind BLAS Level-2 GEMV and the building block of every iterative linear solver.

matrix-vector-product linear-algebra

Metropolis acceptance probability

Paccept=min⁡ ⁣(1,e−β ΔE)P_{\text{accept}} = \min\!\left(1, e^{-\beta\,\Delta E}\right)

Detailed-balance condition for Metropolis Monte Carlo: always accept downhill moves, accept uphill moves with the Boltzmann factor.

metropolis-acceptance statistical-mechanicsalgorithms

Molecular electronic Hamiltonian

H^=−12∑i∇i2−∑i,AZAriA+∑i<j1rij+∑A<BZAZBRAB\hat H = -\tfrac{1}{2}\sum_{i}\nabla_i^{2} - \sum_{i,A}\frac{Z_A}{r_{iA}} + \sum_{i<j}\frac{1}{r_{ij}} + \sum_{A<B}\frac{Z_A Z_B}{R_{AB}}

Born–Oppenheimer electronic Hamiltonian in atomic units. Kinetic energy, electron–nucleus attraction, electron–electron repulsion, and nuclear repulsion — every method in quantum chemistry starts here.

electronic-hamiltonian quantum-chemistry

Monte Carlo error scaling

Error∼σN\text{Error}\sim \frac{\sigma}{\sqrt N}

Statistical error of any Monte Carlo estimate decays as N^{-1/2}. Sets the iconic "halve the error costs four times the work" rule.

mc-error-scaling numerical-methodsalgorithms

Monte Carlo estimate of π

π≈4 points inside circletotal points\pi \approx 4\,\frac{\text{points inside circle}}{\text{total points}}

Classic dart-throwing estimator: ratio of points landing inside the unit-quarter circle scales linearly with π. Toy benchmark for sampling and parallelism.

mc-pi-estimator numerical-methodsalgorithms

Monte Carlo integration

∫abf(x) dx≈b−aN∑i=1Nf(xi),Error∼σN\int_a^b f(x)\,dx \approx \frac{b-a}{N}\sum_{i=1}^{N} f(x_i),\qquad \text{Error}\sim \frac{\sigma}{\sqrt{N}}

Estimate an integral by sampling the integrand at random points. Convergence is dimension-independent at the cost of slow N^{-1/2} accuracy.

monte-carlo-integration numerical-methodsfinance

Monte Carlo option price

V0=e−rT 1N∑i=1NPayoffiV_0 = e^{-rT}\,\frac{1}{N}\sum_{i=1}^{N}\text{Payoff}_i

Risk-neutral price as a discounted sample mean of simulated payoffs. The fallback when no closed form exists, especially for path-dependent or high-dimensional options.

monte-carlo-option-price financenumerical-methods

Monte Carlo option price

V0=e−rT 1N∑i=1NPayoffiV_0 = e^{-rT}\,\frac{1}{N}\sum_{i=1}^{N}\text{Payoff}_i

Risk-neutral expectation of the payoff, discounted to present value and estimated as a simple average over N path simulations.

monte-carlo-discount financealgorithms

Multipole expansion of 1/r12

V(∣r1−r2∣)=∑k=0∞Vk(r1,r2)∑m=−kkYkm(r^1) Ykm∗(r^2)V(|\mathbf r_1 - \mathbf r_2|) = \sum_{k=0}^{\infty} V_k(r_1, r_2)\sum_{m=-k}^{k} Y_{km}(\hat r_1)\,Y^*_{km}(\hat r_2)

Spherical-harmonic addition theorem applied to 1/r12. Separates radial Slater integrals from purely angular Clebsch–Gordan / 6j algebra in the shell model.

multipole-expansion-coulomb nuclear-physicsquantum-chemistry

Newton's method

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Quadratically convergent iteration for finding roots of a smooth scalar function. Generalizes to multi-variable systems via the Jacobian.

newton-method numerical-methodsoptimization

Newton's method update

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Newton–Raphson root finder: locally quadratic convergence when f' is non-zero and the starting guess is close enough.

newton-update numerical-methodsalgorithms

Numerov method

wn+1−2wn+wn−1=h2(gnyn+sn),wn=yn ⁣(1−h212gn)−h212snw_{n+1} - 2w_n + w_{n-1} = h^2\bigl(g_n y_n + s_n\bigr),\quad w_n = y_n\!\left(1 - \tfrac{h^2}{12}g_n\right) - \tfrac{h^2}{12}s_n

Sixth-order accurate three-point recurrence for second-order ODEs of the form y'' = g(x)y + s(x). The classic choice for radial Schrödinger equations.

numerov-method numerical-methodsdifferential-equationsquantum-mechanics

Numerov recover y from w

yn=wn+h212sn1−h212gny_n = \frac{w_n + \tfrac{h^2}{12} s_n}{1 - \tfrac{h^2}{12} g_n}

Algebraic relation that converts Numerov's auxiliary array w_n back to the physical solution y_n at every grid point.

numerov-recovery numerical-methodsdifferential-equations

Numerov stencil

wn+1−2wn+wn−1=h2(gnyn+sn)+O(h6)w_{n+1} - 2w_n + w_{n-1} = h^2(g_n y_n + s_n) + O(h^6)

Three-term recurrence at the heart of Numerov's method for y'' = g(x)y + s(x). Achieves O(h^6) accuracy by absorbing the leading truncation error into the auxiliary variable w.

numerov-stencil numerical-methodsdifferential-equations

ODE initial-value problem

dydt=f(t,y),y(t0)=y0\frac{dy}{dt} = f(t, y), \quad y(t_0) = y_0

Canonical first-order initial-value problem. Every standard ODE solver is built around this signature.

ode-ivp-standard differential-equations

OLS intercept

b=yˉ−m xˉb = \bar y - m\,\bar x

Closed-form intercept for univariate least-squares: passes through the mean of x and y once the slope is fixed.

normal-equations-intercept linear-algebraoptimization

OLS slope

m=n∑xiyi−∑xi∑yin∑xi2−(∑xi)2m = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}

Closed-form least-squares slope for one-dimensional linear regression. Equivalent to Cov(x,y)/Var(x).

normal-equations-slope linear-algebraoptimization

One-electron core operator

h^(i)=−12∇i2−∑AZAriA\hat h(i) = -\tfrac{1}{2}\nabla_i^2 - \sum_A \frac{Z_A}{r_{iA}}

Sum of kinetic energy and electron–nucleus attraction for a single electron — the non-interacting building block of the molecular electronic Hamiltonian.

one-electron-core-operator quantum-chemistry

Operator matrix element

Mij=⟨ϕi∣O^∣ϕj⟩=∫ϕi∗(r) [O^ϕj](r) drM_{ij} = \langle \phi_i | \hat O | \phi_j \rangle = \int \phi_i^*(\mathbf r)\,[\hat O\phi_j](\mathbf r)\,d\mathbf r

Generic Dirac-notation matrix element written as a configuration-space integral; the everyday object of basis-set quantum mechanics.

matrix-element-bracket quantum-mechanicslinear-algebra

Orthogonal map preserves norm

∥Qx∥=∥x∥\|Q x\| = \|x\|

Defining geometric property of orthogonal matrices Q (Q^T Q = I). Why QR factorization is numerically stable for least-squares.

orthogonal-preserves-norm linear-algebra

Orthogonality condition

QTQ=IQ^T Q = I

Defining algebraic property of an orthogonal matrix: columns are orthonormal, so Q^{-1} = Q^T.

orthogonal-condition linear-algebra

Particle in a box energies

En=n2π2ℏ22mL2,ψn(x)=2L sin⁡ ⁣(nπxL)E_n = \frac{n^{2}\pi^{2}\hbar^{2}}{2 m L^{2}},\qquad \psi_n(x) = \sqrt{\tfrac{2}{L}}\,\sin\!\left(\tfrac{n\pi x}{L}\right)

Stationary states of a non-relativistic particle in a 1D infinite square well. The textbook eigenvalue problem and the simplest illustration of energy quantisation.

particle-in-a-box quantum-mechanics

Picard integral equation

y(x)=y(x0)+∫x0xf(t,y(t)) dty(x) = y(x_0) + \int_{x_0}^{x} f(t, y(t))\,dt

Integral form of an ODE initial-value problem. Picard iteration solves it by repeated substitution and converges for Lipschitz f.

picard-integral-form differential-equationsnumerical-methods

Picard iteration

yn+1(x)=y0+∫x0xf(t, yn(t)) dty_{n+1}(x) = y_0 + \int_{x_0}^{x} f\bigl(t,\, y_n(t)\bigr)\,dt

Fixed-point iteration on the integral form of an IVP. Underpins the Picard–Lindelöf existence-and-uniqueness theorem for ODEs.

picard-iteration differential-equationsnumerical-methods

Plane wave in spherical waves

eik⋅r=4π∑ℓ=0∞∑m=−ℓℓiℓjℓ(kr) Yℓm∗(k^) Yℓm(r^)e^{i\mathbf k\cdot\mathbf r} = 4\pi\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} i^\ell j_\ell(kr)\,Y_\ell^{m*}(\hat{\mathbf k})\,Y_\ell^m(\hat{\mathbf r})

Rayleigh expansion of a plane wave in spherical Bessel functions and spherical harmonics. The bridge between plane-wave and atomic-sphere descriptions in APW/LAPW methods.

plane-wave-spherical-expansion solid-state-physicsquantum-mechanicsseries

Power iteration

vk+1=A vk∥A vk∥,λ≈vkTA vkvkTvk\mathbf v_{k+1} = \frac{A\,\mathbf v_k}{\lVert A\,\mathbf v_k\rVert},\qquad \lambda \approx \frac{\mathbf v_k^{T} A\,\mathbf v_k}{\mathbf v_k^{T}\mathbf v_k}

Repeated multiplication by A with renormalisation converges to the dominant eigenvector. The simplest member of the Krylov-iteration family and the seed of Arnoldi and Lanczos.

power-iteration linear-algebraeigenvalue

Power iteration step

vk+1=Avk∥Avk∥\mathbf v_{k+1} = \frac{A\mathbf v_k}{\|A\mathbf v_k\|}

Rescales A v to unit norm at every step. Converges to the dominant eigenvector at rate |λ_2|/|λ_1|.

power-iteration-update linear-algebraeigenvaluealgorithms

Power-method neural-network loss

LPMNN(θ)=1N∑i=1N[Uk−1(xi)−Uk(xi)]2\mathcal L_{\text{PMNN}}(\theta) = \frac{1}{N}\sum_{i=1}^{N}\big[U_{k-1}(x_i) - U_k(x_i)\big]^2

Mean-squared difference between two consecutive iterates in the power-method neural network. Driving the loss to zero realizes convergence of the underlying eigenvector iteration.

pmnn-loss numerical-methodsoptimizationeigenvalue

Put–call parity

C−P=S−Ke−rTC - P = S - K e^{-rT}

Model-free no-arbitrage relation linking European call and put prices with the same strike and maturity. Fix any three of the prices and the fourth follows.

put-call-parity finance

QR factorization via Householder

Q=H1H2⋯HnQ = H_1 H_2 \cdots H_n

Composing Householder reflections produces an orthogonal Q such that QA = R is upper-triangular. The standard route to least-squares solutions and the QR eigenvalue algorithm.

qr-householder-product linear-algebra

QR least-squares back-substitution

Rx=(QTb)1:nR x = (Q^T b)_{1:n}

Final step of QR-based least-squares: apply Q^T to b and triangular-solve against the n×n top block of R.

qr-back-substitution linear-algebraalgorithms

Radial Hamiltonian in atomic sphere

hl=−1rd2dr2r+l(l+1)r2+V(r)h_l = -\frac{1}{r}\frac{d^2}{dr^2}r + \frac{l(l+1)}{r^2} + V(r)

Spherical Hamiltonian acting on radial functions inside a muffin-tin. Solving h_l u_l = E u_l generates the APW/LAPW radial parts.

lapw-radial-hamiltonian solid-state-physicsquantum-mechanics

Radial Schrödinger equation

−1rd2dr2(rul)+l(l+1)r2ul+V(r)ul=Eul-\frac{1}{r}\frac{d^2}{dr^2}(r u_l) + \frac{l(l+1)}{r^2} u_l + V(r) u_l = E u_l

Schrödinger equation reduced to one dimension after separating angular variables; the centrifugal term l(l+1)/r^2 is the angular-momentum barrier.

radial-schrodinger quantum-mechanics

Rayleigh plane-wave expansion

eik⋅r=4π∑ℓ=0∞∑m=−ℓℓiℓ jℓ(kr) Yℓm(r^) Yℓm∗(k^)e^{i\mathbf k\cdot\mathbf r} = 4\pi\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} i^{\ell}\,j_{\ell}(k r)\,Y_{\ell}^{m}(\hat{\mathbf r})\,Y_{\ell}^{m*}(\hat{\mathbf k})

Decomposition of a plane wave into spherical Bessel functions and spherical harmonics. The matching trick that connects interstitial plane waves to muffin-tin partial waves in APW/LAPW methods.

rayleigh-plane-wave solid-state-physicsseries

Rayleigh quotient

R(v)=vTA vvTvR(\mathbf v) = \frac{\mathbf v^{T} A\,\mathbf v}{\mathbf v^{T}\mathbf v}

Stationary points of the Rayleigh quotient are eigenvectors and the values are eigenvalues. The variational principle behind power iteration, Lanczos, and electronic-structure methods.

rayleigh-quotient linear-algebraeigenvalue

Real Fourier series expansion

f(x)∼a02+∑n=1∞[ancos⁡ ⁣(2nπxP)+bnsin⁡ ⁣(2nπxP)]f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\!\left(\frac{2 n \pi x}{P}\right) + b_n\sin\!\left(\frac{2 n \pi x}{P}\right)\right]

Standard real-valued Fourier expansion of a P-periodic function. The basis is mutually orthogonal on [0,P] and the expansion converges in L^2.

fourier-series-real series

Real-space lattice vector

Rn=n1a1+n2a2+n3a3\mathbf R_n = n_1\mathbf a_1 + n_2\mathbf a_2 + n_3\mathbf a_3

A general translation vector of a Bravais lattice — integer combinations of the three primitive vectors a_i.

real-space-lattice-vector solid-state-physics

Reciprocal lattice vectors

Gm=m1b1+m2b2+m3b3\mathbf G_m = m_1\mathbf b_1 + m_2\mathbf b_2 + m_3\mathbf b_3

Dual lattice in momentum space whose primitive vectors satisfy a_i·b_j = 2π δ_{ij}. Defines Brillouin zones, Bloch states, and diffraction conditions.

reciprocal-lattice solid-state-physics

Reciprocal-lattice vector

Gm=m1b1+m2b2+m3b3\mathbf G_m = m_1\mathbf b_1 + m_2\mathbf b_2 + m_3\mathbf b_3

General reciprocal-lattice vector built from integer combinations of the primitive reciprocal vectors. Indexes the diffraction conditions and Fourier components of periodic functions.

reciprocal-vector-expansion solid-state-physics

Roothaan–Hall equations

F C=S C ϵ\mathbf F\,\mathbf C = \mathbf S\,\mathbf C\,\boldsymbol\epsilon

Generalized eigenvalue problem solved at every step of the Hartree–Fock SCF loop. F is the Fock matrix, S the overlap matrix.

hartree-fock-roothaan quantum-chemistryhartree-fock

Runge–Kutta RK4 stages

k1=f(tn,yn)k2=f(tn+h/2, yn+hk1/2)k3=f(tn+h/2, yn+hk2/2)k4=f(tn+h, yn+hk3)yn+1=yn+h6(k1+2k2+2k3+k4)\begin{aligned} k_1 &= f(t_n, y_n) \\ k_2 &= f(t_n + h/2,\, y_n + h k_1/2) \\ k_3 &= f(t_n + h/2,\, y_n + h k_2/2) \\ k_4 &= f(t_n + h,\, y_n + h k_3) \\ y_{n+1} &= y_n + \tfrac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4) \end{aligned}

Stage equations of the classic 4th-order Runge–Kutta method. Local truncation error O(h^5), global O(h^4).

rk4-stages differential-equationsnumerical-methods

Sample covariance

Cov(X,Y)=1n∑i=1n(xi−xˉ)(yi−yˉ)\text{Cov}(X,Y) = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar x)(y_i - \bar y)

Mean-centred dot product divided by n. Off-diagonal entry of the sample covariance matrix.

covariance-formula linear-algebra

Shell-model two-body matrix element

⟨ab∣V∣cd⟩=∑k(Clebsch–Gordan / 6j / 9j algebra)×Fk(ab,cd)\langle ab | V | cd \rangle = \sum_k \big(\text{Clebsch–Gordan / 6j / 9j algebra}\big)\times F^k(ab,cd)

Schematic statement that nuclear two-body matrix elements factor into angular-momentum coupling coefficients times Slater radial integrals.

shell-model-two-body nuclear-physics

Single-particle level degeneracy

gl=2(2l+1)g_l = 2(2l+1)

Number of magnetic sublevels (with spin) accommodated by a single-particle orbital of angular momentum l. Used to fill shells and locate magic numbers.

nuclear-shell-degeneracy nuclear-physics

Slater determinant

∣Φ⟩=1N! det⁡ ⁣[χi(j)]i,j=1N|\Phi\rangle = \frac{1}{\sqrt{N!}}\,\det\!\bigl[\chi_i(j)\bigr]_{i,j=1}^{N}

Antisymmetrized product of one-electron spin-orbitals. The minimal wavefunction that satisfies the Pauli principle and the building block of Hartree–Fock and post-HF expansions.

slater-determinant quantum-chemistry

Slater determinant (A nucleons)

Ψ(1,2,…,A)=1A!det⁡(ϕ1(1)ϕ2(1)⋯ϕ1(2)ϕ2(2)⋯⋮⋮⋱)\Psi(1,2,\ldots,A) = \frac{1}{\sqrt{A!}}\det\begin{pmatrix} \phi_1(1) & \phi_2(1) & \cdots \\ \phi_1(2) & \phi_2(2) & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}

Antisymmetric many-body wavefunction built from single-particle orbitals. The starting point of independent-particle models in nuclear and electronic structure.

slater-determinant-nuclear quantum-chemistrynuclear-physics

Slater radial integral F^k

Fk(ab,cd)=∫dr1 r12∫dr2 r22 Ra(r1)Rc(r1) Vk(r1,r2) Rb(r2)Rd(r2)F^k(ab,cd) = \int dr_1\,r_1^2 \int dr_2\,r_2^2\,R_a(r_1)R_c(r_1)\,V_k(r_1,r_2)\,R_b(r_2)R_d(r_2)

Reusable radial part of a multipole-expanded two-body matrix element. All angular dependence is handled separately by 3j/6j coefficients.

slater-radial-integral nuclear-physicsquantum-chemistry

Slater-type orbital (spherical)

ϕ(r)=rn−1e−ζr Ylm(θ,ϕ)\phi(\mathbf r) = r^{n-1} e^{-\zeta r}\,Y_{lm}(\theta,\phi)

Standard radial × spherical-harmonic factorization of a Slater-type orbital. Captures the cusp at the nucleus exactly, unlike Gaussians.

sto-spherical quantum-chemistry

Slater–Condon diagonal element

HII=∑ihii+∑i<j[(ii∣jj)−(ij∣ji)]H_{II} = \sum_i h_{ii} + \sum_{i<j} \big[(ii|jj) - (ij|ji)\big]

Energy expectation value for a single Slater determinant: sum of orbital one-electron energies plus pairwise direct minus exchange Coulomb terms.

slater-condon-diagonal quantum-chemistry

Spherical harmonic in (θ, φ)

Ylm(θ,ϕ)∝Plm(cos⁡θ) eimϕY_l^m(\theta,\phi) \propto P_l^m(\cos\theta)\,e^{im\phi}

Standard angular eigenfunction of L^2 and L_z, factorized into associated-Legendre polynomial in cos θ times an azimuthal phase.

spherical-harmonics-form quantum-mechanicsseries

Spin–orbit interaction

VLS=−VLS l⋅sV_{LS} = -V_{LS}\,\mathbf l\cdot\mathbf s

Coupling between orbital and spin angular momentum, responsible for the j = l ± 1/2 splitting and the large gaps that produce nuclear magic numbers.

spin-orbit-interaction nuclear-physicsquantum-mechanics

STO-nG contraction

ψSTO(r)≈∑i=1nci ⁣(2αiπ)3/4e−αir2\psi_{\text{STO}}(r) \approx \sum_{i=1}^{n} c_i\!\left(\tfrac{2\alpha_i}{\pi}\right)^{3/4} e^{-\alpha_i r^{2}}

Pople-style fit of a Slater-type orbital by a fixed contraction of n Gaussians. Recovers the qualitative behavior of an STO while keeping integrals analytic.

sto-ng-fit quantum-chemistry

STO-nG fit objective

R(p)=∑j[ψSTO-nG(rj;p)−ψSTO(rj)]2R(\mathbf p) = \sum_j \big[\psi_{\text{STO-nG}}(r_j;\mathbf p) - \psi_{\text{STO}}(r_j)\big]^2

Sum-of-squares residual minimized when fitting a contracted-Gaussian basis to a reference Slater orbital on a radial grid.

sto-ng-objective quantum-chemistryoptimization

Susceptibility critical scaling

χ∼∣T−Tc∣−γ\chi \sim |T - T_c|^{-\gamma}

Magnetic susceptibility diverges with critical exponent γ near T_c. γ = 7/4 in the 2D Ising model.

critical-susceptibility statistical-mechanics

Symmetric orthogonalisation

X=S−1/2X = S^{-1/2}

Löwdin's symmetric transformation that turns a non-orthogonal AO basis into an orthogonal one. Lets the generalized eigenvalue problem be reduced to a standard one.

orthogonalising-transformation quantum-chemistrylinear-algebra

Symplectic Euler for harmonic oscillator

vn+1=vn−Δt ω2yn,yn+1=yn+Δt vn+1v_{n+1} = v_n - \Delta t\,\omega^2 y_n, \quad y_{n+1} = y_n + \Delta t\,v_{n+1}

Order-1 symplectic integrator for y'' + ω² y = 0. Conserves a discrete energy almost exactly, unlike forward Euler.

verlet-half-step differential-equationsnumerical-methods

Time-dependent Schrödinger equation

iℏ ∂∂t∣Ψ(t)⟩=H^ ∣Ψ(t)⟩i\hbar\,\frac{\partial}{\partial t}|\Psi(t)\rangle = \hat H\,|\Psi(t)\rangle

Governs how a quantum state evolves in time under a (possibly time-dependent) Hamiltonian.

schrodinger-tdse quantum-mechanicsdynamics

Time-independent Schrödinger equation

H^ ∣ψ⟩=E ∣ψ⟩\hat H \,|\psi\rangle = E\,|\psi\rangle

Eigenvalue equation for stationary states of the Hamiltonian. The cornerstone of non-relativistic quantum mechanics.

schrodinger-tise quantum-mechanicseigenvalue

Transformed Fock matrix

H~=XTHX\tilde H = X^T H X

Fock matrix in the orthogonal basis obtained by sandwiching with X = S^{-1/2}. Diagonalizing it gives orbital energies directly.

transformed-fock-matrix quantum-chemistrylinear-algebra

Trapezoidal rule

∫abf(x) dx≈h2[f(a)+2∑i=1N−1f(a+ih)+f(b)]\int_a^b f(x)\,dx \approx \frac{h}{2}\left[f(a) + 2\sum_{i=1}^{N-1} f(a + ih) + f(b)\right]

Composite quadrature rule built from straight-line interpolants on a uniform grid. Second-order accurate and the baseline against which fancier integrators are measured.

trapezoidal-rule numerical-methods

Trapezoidal rule error

E=−(b−a)312N2 f′′(ξ)E = -\frac{(b-a)^{3}}{12 N^{2}}\,f''(\xi)

Composite-trapezoidal error term, scaling like 1/N² and proportional to the second derivative. Vanishes for linear integrands and motivates Simpson and Romberg refinements.

trapezoidal-error numerical-methods

Two-electron integral (chemist notation)

(μν∣λσ)=∬ϕμ(r1)ϕν(r1) 1∣r1−r2∣ ϕλ(r2)ϕσ(r2) dr1 dr2(\mu\nu | \lambda\sigma) = \iint \phi_\mu(\mathbf r_1)\phi_\nu(\mathbf r_1)\,\frac{1}{|\mathbf r_1 - \mathbf r_2|}\,\phi_\lambda(\mathbf r_2)\phi_\sigma(\mathbf r_2)\,d\mathbf r_1\,d\mathbf r_2

Repulsion integral over four basis functions in chemist's (11|22) ordering. The scaling bottleneck (M^4) of conventional Hartree–Fock and post-HF methods.

two-electron-integral-chemist quantum-chemistry

Two-electron repulsion integral

(μν ∣ λσ)=∬ϕμ(r1)ϕν(r1) 1r12 ϕλ(r2)ϕσ(r2) dr1 dr2(\mu\nu\,|\,\lambda\sigma) = \iint \phi_{\mu}(\mathbf r_1)\phi_{\nu}(\mathbf r_1)\,\frac{1}{r_{12}}\,\phi_{\lambda}(\mathbf r_2)\phi_{\sigma}(\mathbf r_2)\,d\mathbf r_1\,d\mathbf r_2

Coulomb-repulsion integrals over four basis functions. Their evaluation and storage dominate the cost of every Gaussian-basis quantum-chemistry program.

two-electron-integral quantum-chemistry

Two-electron singlet state

∣Ψsinglet⟩=12(∣1↑2↓⟩−∣1↓2↑⟩)|\Psi_{\text{singlet}}\rangle = \tfrac{1}{\sqrt{2}}\big(|1_\uparrow 2_\downarrow\rangle - |1_\downarrow 2_\uparrow\rangle\big)

Antisymmetric (S=0, S_z=0) spin combination of two electrons in distinct spatial orbitals. The minimal example of entangled spin pair.

singlet-two-electron quantum-chemistryquantum-mechanics

Two-site Hubbard Hamiltonian

H^=−t∑σ(c^1σ†c^2σ+c^2σ†c^1σ)+U∑i=12n^i↑n^i↓\hat H = -t\sum_{\sigma}\big(\hat c_{1\sigma}^\dagger \hat c_{2\sigma} + \hat c_{2\sigma}^\dagger \hat c_{1\sigma}\big) + U\sum_{i=1}^{2} \hat n_{i\uparrow} \hat n_{i\downarrow}

Minimal lattice model with kinetic hopping t and on-site repulsion U. Captures the competition between delocalisation and Coulomb correlation in two orbitals.

hubbard-dimer-hamiltonian quantum-chemistrysolid-state-physics

Variational principle

E0  ≤  ⟨Ψtrial∣H^∣Ψtrial⟩⟨Ψtrial∣Ψtrial⟩E_0 \;\leq\; \frac{\langle\Psi_{\text{trial}}|\hat H|\Psi_{\text{trial}}\rangle}{\langle\Psi_{\text{trial}}|\Psi_{\text{trial}}\rangle}

Any normalizable trial wavefunction gives an upper bound on the ground-state energy. The justification for Hartree–Fock, configuration interaction, and the entire variational toolbox.

variational-principle quantum-chemistryquantum-mechanics

Variational principle upper bound

E0≤⟨ψ∣H^∣ψ⟩⟨ψ∣ψ⟩E_0 \le \frac{\langle\psi|\hat H|\psi\rangle}{\langle\psi|\psi\rangle}

Rayleigh–Ritz inequality: the expectation value of H in any trial state bounds the true ground-state energy from above. Backbone of every variational method.

variational-upper-bound quantum-mechanicsquantum-chemistry

Vector dot product

a⋅b=∑i=1naibi\mathbf a\cdot\mathbf b = \sum_{i=1}^{n} a_i b_i

Sum of pairwise products — the canonical BLAS Level-1 dot routine. Foundation of inner products, norms, and projections.

dot-product linear-algebra

Woods–Saxon potential

V(r)=−V01+exp⁡ ⁣(r−Ra)V(r) = -\frac{V_0}{1 + \exp\!\left(\dfrac{r - R}{a}\right)}

Smoothed mean-field potential approximating the nuclear interior. Used together with a spin-orbit term to reproduce the magic numbers in the nuclear shell model.

woods-saxon nuclear-physics