Glossary

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2D finite difference finite-difference-2d

Discretization of the 2D Laplacian on a regular grid via the five-point stencil (u_{i±1,j} + u_{i,j±1} − 4 u_{ij})/h². Yields a sparse, banded linear system for Poisson and Laplace equations.

A* search a-star

Best-first graph search that orders nodes by f(n) = g(n) + h(n), where g is path cost so far and h is an admissible heuristic estimating remaining cost. Optimal and complete with admissible h.

Adomian decomposition adomian-decomposition

A semi-analytic series method for nonlinear functional equations, expanding the solution and the nonlinearity in special Adomian polynomials. Yields recursion relations that can be summed or Padé-accelerated.

Amortized analysis amortized-analysis

A technique for averaging the cost of a sequence of operations on a data structure, even when individual operations are occasionally expensive. Common examples: dynamic arrays and union–find.

Anderson acceleration anderson-acceleration

A convergence accelerator that combines a few past iterates of a fixed-point map by minimizing the residual of their linear combination. Widely used to speed up SCF and Picard iterations.

Annihilation operator annihilation-operator

The second-quantized operator â_p that removes a fermion from spin orbital p, satisfying anticommutation with creation operators. Acts as zero on the vacuum.

Antisymmetry antisymmetry

The requirement that a many-fermion wavefunction change sign under the exchange of any two particle labels. Imposed via Slater determinants in molecular and atomic problems.

Arnoldi iteration arnoldi-iteration

A Krylov-subspace method that builds an orthonormal basis for span{v, Av, A^2 v, ...} and projects A onto it as an upper Hessenberg matrix. Used for eigenvalues of large nonsymmetric matrices.

Associated Legendre polynomials associated-legendre-polynomials

Solutions P_l^m(x) of the associated Legendre equation; combined with e^{imφ} they form spherical harmonics. They appear in any problem with axial symmetry.

Atomic orbital atomic-orbital

A one-electron wavefunction associated with a single atomic center, used as a building block in LCAO expansions of molecular orbitals. Common forms are Slater-type and Gaussian-type orbitals.

Augmented plane wave method apw-method

A basis for solid-state electronic-structure calculations that uses plane waves in the interstitial region and atom-centered radial solutions of the Schrödinger equation inside spheres around each nucleus, matched at the boundary.

Band structure band-structure

The set of energy bands E_n(k) of electrons in a periodic potential, plotted along high-symmetry directions in the Brillouin zone. Determines metallicity, gaps, and effective masses.

Basis set basis-set

A finite collection of one-electron functions used to expand molecular orbitals or many-electron wavefunctions. The basis controls accuracy and cost; common families include STO-nG, Pople, and correlation-consistent sets.

Bellman–Ford algorithm bellman-ford

Single-source shortest paths in O(VE) that handles negative edge weights and detects negative cycles. Slower than Dijkstra but more general.

Big-O notation big-o-notation

Asymptotic upper bound f(n) = O(g(n)) meaning f grows at most as fast as a constant multiple of g for large n. Standard tool for stating algorithm time and space complexity.

Binary search binary-search

A divide-and-conquer search that locates a target in a sorted array in O(log n) by repeatedly halving the search interval. Precondition: the array must be sorted.

Binder cumulant binder-cumulant

The dimensionless ratio U_4 = 1 − ⟨m^4⟩/(3⟨m^2⟩²) used in finite-size scaling of phase transitions. Its crossing of curves for different system sizes locates the critical temperature.

Bisection method bisection-method

A root-finding algorithm that brackets a sign change of a continuous function and repeatedly halves the interval. Convergence is linear but guaranteed.

Black–Scholes model black-scholes-model

A continuous-time model of European option pricing in which the underlying follows geometric Brownian motion. Its closed-form solution gives the no-arbitrage price as a function of spot, strike, rate, time, and volatility.

BLAS blas

Basic Linear Algebra Subprograms: a tiered API (Level 1–3) for vector–vector, matrix–vector, and matrix–matrix kernels like DAXPY and DGEMM. Performance-tuned BLAS underlies almost all numerical software.

Bloch's theorem bloch-theorem

States that eigenfunctions of a Hamiltonian periodic in a Bravais lattice can be written as e^{ik·r} u_k(r) with u_k periodic. It gives the band-structure framework of solid-state physics.

Block Davidson method block-davidson

A generalization of the Davidson algorithm that expands the subspace by several correction vectors per iteration, targeting multiple eigenpairs simultaneously. Standard for low-lying eigenstates of large CI Hamiltonians.

Bond dissociation bond-dissociation

The process of pulling a chemical bond apart toward separated fragments. Restricted Hartree–Fock fails qualitatively in this limit, motivating multireference and FCI treatments.

Born–Oppenheimer approximation born-oppenheimer-approximation

Separates electronic and nuclear motion by treating nuclei as fixed point charges while solving the electronic Schrödinger equation. It is the foundation of nearly all molecular electronic-structure theory.

Boundary matching boundary-matching

In APW/LAPW, the requirement that wavefunctions and their derivatives agree across the muffin-tin boundary. Sets the linear coefficients that build the augmented basis function.

Boyer–Moore algorithm boyer-moore

A string-matching algorithm that scans the pattern right-to-left and uses bad-character and good-suffix heuristics to skip large portions of text. Often the fastest in practice for long patterns.

Boys function boys-function

The auxiliary function F_0(t) = ½√(π/t) erf(√t) that appears whenever Gaussian primitives are combined to evaluate Coulomb-type integrals. It is the workhorse of analytic two-electron integral evaluation.

Breadth-first search breadth-first-search

Graph traversal that visits all vertices at the current distance before going deeper, using a FIFO queue. Yields shortest paths in unweighted graphs.

Brent's method brent-method

A robust hybrid root-finder combining bisection, secant, and inverse quadratic interpolation. Guarantees convergence on a bracketed root with near-quadratic speed when smooth.

Brillouin zone brillouin-zone

The Wigner–Seitz cell of the reciprocal lattice; the natural domain for crystal momentum k. Bloch states with k differing by a reciprocal-lattice vector are equivalent, so band structures are plotted only over one zone.

Cesàro summation cesaro-summation

A regularization that assigns a value to a divergent series via the limit of the average of its partial sums. Used in Fourier analysis (Fejér summation) and to relate Picard iterates to Adomian series.

Chebyshev polynomials chebyshev-polynomials

The orthogonal polynomials on [-1, 1] with weight 1/sqrt(1 - x^2), defined by T_n(cos theta) = cos(n theta). Their roots are the optimal nodes for polynomial interpolation and underlie spectral collocation methods.

Collocation method collocation-method

A spectral approach that enforces a differential equation exactly at chosen collocation points using a global polynomial expansion. Chebyshev–Lobatto points give spectrally accurate, well-conditioned schemes.

Condition number condition-number

For a matrix A, κ(A) = ‖A‖ · ‖A^{-1}‖, measuring how much output errors can be amplified relative to input errors. Small κ means well-conditioned; large κ means ill-conditioned.

Configuration interaction configuration-interaction

A post-Hartree–Fock method that expands the wavefunction as a linear combination of Slater determinants and diagonalizes the many-body Hamiltonian in that basis. CISD truncates at single and double excitations; FCI keeps all of them.

Contraction coefficient contraction-coefficient

The fixed coefficient c_i multiplying each primitive Gaussian when forming a contracted basis function. Together with exponents it defines a contracted Gaussian-type orbital.

Core Hamiltonian core-hamiltonian

The one-electron part of the Fock matrix, summing kinetic energy and electron–nucleus attraction integrals over basis functions. It is independent of the density and computed once per geometry.

Correction vector correction-vector

In Davidson’s method, the preconditioned residual t = (μI − D)^{-1} r used to expand the search subspace each iteration. Choice of preconditioner determines convergence rate.

Coulomb operator coulomb-operator

The Hartree–Fock mean-field operator Ĵ_j(1) representing the classical electrostatic repulsion of electron 1 from the charge cloud of orbital j. Its matrix element with another orbital is a two-electron integral.

Coupled cluster coupled-cluster

A post-Hartree–Fock method that parameterizes the wavefunction as e^T |Φ_HF⟩ with a cluster excitation operator T. Truncations like CCSD and CCSD(T) provide highly accurate, size-extensive correlation energies.

Covariance matrix covariance-matrix

The symmetric positive-semidefinite matrix Σ_{ij} = E[(X_i − μ_i)(X_j − μ_j)] of pairwise covariances. Eigenvectors give principal components.

Creation operator creation-operator

The second-quantized operator â^†_p that adds a fermion to spin orbital p, with anticommutation rules enforcing antisymmetry. Together with annihilation operators it builds every many-body operator.

Critical exponents critical-exponents

Universal power-law exponents (β, γ, ν, …) governing the divergence of thermodynamic quantities near a continuous phase transition. They are organized into universality classes.

Crystal lattice crystal-lattice

A periodic arrangement of points in space generated by integer combinations of a set of primitive vectors. Together with a basis of atoms, it defines a crystal structure.

Crystal structure crystal-structure

A periodic arrangement of atoms described by a Bravais lattice plus a basis. Determines the symmetry properties exploited by Bloch states and band-structure calculations.

Davidson method davidson-method

A subspace iteration for the lowest few eigenpairs of a large Hermitian matrix, augmenting the search subspace with preconditioned residual vectors. The method of choice for large-scale CI and TDDFT eigenproblems.

DAXPY daxpy

BLAS Level-1 kernel performing y ← α x + y in double precision. Canonical example of a streaming, bandwidth-bound numerical loop.

Delta (Greek) delta

The partial derivative ∂V/∂S of an option value with respect to the underlying spot. For a European call under Black–Scholes, delta = N(d_1).

Density functional theory density-functional-theory

A reformulation of the many-electron problem in terms of the electron density rather than the wavefunction, justified by the Hohenberg–Kohn theorems. In practice the density is obtained by solving the Kohn–Sham single-particle equations with an approximate exchange–correlation functional.

Density matrix density-matrix

In Hartree–Fock theory, P_{μν} = 2 Σ_i C_{μi} C^*_{νi} summed over occupied orbitals, encoding the one-electron density in the AO basis. SCF iterates the density to self-consistency.

Depth-first search depth-first-search

Graph traversal that explores as far as possible along each branch before backtracking, using a stack (or recursion). Underpins topological sort, SCC decomposition, and cycle detection.

Differentiation matrix differentiation-matrix

A dense matrix D such that (Df)_i approximates f'(x_i) at collocation nodes. The Chebyshev differentiation matrix gives spectral accuracy for smooth functions.

DIIS diis

Direct Inversion in the Iterative Subspace: Pulay’s method for accelerating SCF by extrapolating the Fock matrix from past iterates that minimize an error vector. Standard tool in quantum chemistry.

Dijkstra's algorithm dijkstra-algorithm

Single-source shortest paths in a weighted graph with non-negative edge weights, using a priority queue to greedily extract the closest unsettled vertex. Runs in O((V + E) log V) with a binary heap.

Divide-and-conquer divide-and-conquer

An algorithmic paradigm that splits a problem into smaller subproblems, solves each recursively, and combines results. Powers merge sort, FFT, Strassen multiplication, and many spectral solvers.

Dynamic programming dynamic-programming

An algorithmic technique for problems with overlapping subproblems and optimal substructure: solve each subproblem once and cache the result. The basis of the standard solutions to knapsack, edit distance, and longest common subsequence.

Eigenvalue eigenvalue

A scalar λ such that for some operator A and non-zero vector v, Av = λv. The vector v is called an eigenvector.

Electron affinity electron-affinity

The energy released when an extra electron is added to a neutral atom or molecule. Approximated by minus the LUMO energy in single-determinant theory.

Electron correlation electron-correlation

The energy difference E_exact − E_HF, capturing the correlated motion of electrons that mean-field Hartree–Fock cannot describe. Recovering it is the central task of post-HF methods.

Electron–nucleus integral electron-nucleus-integral

A one-electron matrix element ⟨φ_μ| -Z/|r-R_N| |φ_ν⟩ giving the attraction of an electron to a nucleus. With Gaussian primitives it reduces to the Boys F_0 function.

ERI tensor eri-tensor

The four-index tensor of electron-repulsion integrals (μν|λσ) used to assemble the Coulomb and exchange contributions of the Fock matrix. Storing it scales as N^4 in basis size and dominates HF cost.

Euler method euler-method

The simplest numerical integrator for ordinary differential equations, advancing y by h f(t, y) at each step. It has first-order global accuracy and is the textbook starting point for more sophisticated schemes.

Euler–Maruyama method euler-maruyama

The stochastic analog of explicit Euler for SDEs, advancing dY = a dt + b dW by a · Δt + b · √Δt · ξ with ξ standard normal. First-order strong scheme commonly used in finance simulations.

Exchange operator exchange-operator

The non-local operator K̂_j arising in Hartree–Fock from antisymmetry of the wavefunction; it has no classical analog and lowers the energy of same-spin electron pairs. Together with Ĵ it builds the mean-field part of the Fock operator.

Exchange–correlation functional exchange-correlation-functional

The unknown density functional E_xc[ρ] that contains all many-body effects beyond the classical Coulomb energy in Kohn–Sham DFT. Approximations include LDA, GGA (e.g., PBE), and hybrid functionals.

Excited state excited-state

Any eigenstate of a Hamiltonian with energy above the ground state. Block Davidson and CI methods target several low-lying excited states at once.

Fejér summation fejer-summation

The Cesàro mean of partial Fourier sums; smooths out the Gibbs phenomenon and converges uniformly for continuous periodic functions. A standard tool of classical Fourier analysis.

Fermion fermion

A particle with half-integer spin obeying Fermi–Dirac statistics and the Pauli exclusion principle. Electrons, protons, and neutrons are fermions.

Finite difference method finite-difference-method

A discretization that replaces derivatives by differences of values at neighboring grid points, turning a differential equation into a linear or nonlinear algebraic system. The simplest way to put a PDE on a computer.

Finite element method finite-elements

A discretization that expands the solution of a PDE in piecewise-polynomial basis functions over a mesh and enforces Galerkin or collocation conditions. Standard tool for elliptic and parabolic PDEs in complex geometries.

Finite-difference stencil finite-difference-stencil

A pattern of grid points used to approximate a differential operator by a weighted sum of function values. Stencils encode the order, accuracy, and bandwidth of the resulting discrete operator.

Fixed-point iteration fixed-point-iteration

An iteration of the form x_{k+1} = g(x_k) that converges, when stable, to a solution of x = g(x). The SCF loop and Picard iteration are concrete examples.

Fixed-point theorem fixed-point-theorem

Result guaranteeing existence (and sometimes uniqueness) of x with g(x) = x. Banach’s contraction-mapping theorem also yields convergence of fixed-point iteration.

Floyd–Warshall algorithm floyd-warshall

All-pairs shortest paths via a triple loop in O(V^3), based on dynamic programming over intermediate vertices. Works with negative edges if there are no negative cycles.

Fock matrix fock-matrix

The matrix representation of the Fock operator in an atomic-orbital basis, equal to the core Hamiltonian plus density-dependent Coulomb and exchange contributions. Diagonalizing it yields molecular orbital energies and coefficients.

Fock operator fock-operator

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

Fourier series fourier-series

An expansion of a periodic function as a sum of sines and cosines with frequencies that are integer multiples of the fundamental. The coefficients are inner products of the function against the basis elements.

Full configuration interaction full-configuration-interaction

The exact diagonalization of the electronic Hamiltonian within a given one-electron basis: the wavefunction is expanded in every Slater determinant that can be built from the orbitals. Cost grows factorially with system size.

Gamma (Greek) gamma-greek

The second derivative ∂²V/∂S² of an option value with respect to spot, equivalently ∂Δ/∂S. Measures the convexity of the option position.

Gauss–Seidel iteration gauss-seidel

An iterative linear solver that updates each component of the solution in place using the most recent values of the others. Converges for diagonally dominant or symmetric positive-definite matrices.

Gauss–Seidel iteration gauss-seidel-iteration

A stationary iterative solver for Ax = b that uses the most recent updates within a sweep, accelerating Jacobi convergence at the cost of inherent sequentiality. Converges for diagonally dominant or SPD systems.

Gaussian elimination gaussian-elimination

The standard direct method for solving a linear system, in which row operations reduce the matrix to upper triangular form and the solution is obtained by back substitution. Partial pivoting is normally used for numerical stability.

Gaussian exponent gaussian-exponent

The α parameter controlling the radial decay of a Gaussian primitive e^{-α r²}. Small exponents describe diffuse functions, large exponents describe tight core functions.

Gaussian product theorem gaussian-product-theorem

States that the product of two Gaussian functions centered at different points is itself a Gaussian centered at a weighted midpoint, scaled by an overlap factor. This identity makes integrals over Gaussian primitives analytically tractable.

Gaussian-type orbital gaussian-type-orbital

A basis function of the form exp(-alpha r^2) times a polynomial in the Cartesian coordinates. Two-electron integrals over Gaussians are analytic, which is why nearly all production quantum-chemistry codes use them.

Generalized eigenvalue problem generalized-eigenvalue-problem

The matrix equation A v = λ B v, with B typically symmetric positive-definite. The Roothaan–Hall equations and APW/LAPW secular equations both take this form.

Geometric Brownian motion geometric-brownian-motion

The stochastic process dS = mu S dt + sigma S dW. It is the standard model for asset prices because its logarithm is a Brownian motion with drift and it stays strictly positive.

Gibbs phenomenon gibbs-phenomenon

The fixed ~9% overshoot near a jump discontinuity that persists when truncating a Fourier series, regardless of how many terms are kept. Cesàro/Fejér summation suppresses it.

Givens rotation givens-rotation

A 2×2 plane rotation used to selectively zero entries of a matrix. Alternative to Householder reflections for QR factorization, especially in sparse settings.

Gram–Schmidt process gram-schmidt

A procedure for orthonormalizing a set of vectors by sequentially subtracting projections onto previously processed vectors. Modified Gram–Schmidt is the numerically stable variant used in Arnoldi.

Greedy algorithm greedy-algorithm

A strategy that builds a solution by always making the locally optimal choice, hoping it yields a global optimum. Works for problems with matroid or exchange-argument structure (e.g., MST).

Ground state ground-state

The lowest-energy eigenstate of a Hamiltonian. Variational methods deliver upper bounds on its energy.

Hamiltonian hamiltonian

The energy operator of a quantum system; its eigenvalues are the allowed energies. For an N-electron molecule it sums kinetic, electron–nucleus, and electron–electron contributions.

Harmonic oscillator harmonic-oscillator

A particle in a quadratic potential V(x) = ½ k x², the canonical exactly solvable quantum system with equally spaced energy levels ℏω(n + ½). Used as a test problem for finite-difference and spectral methods.

Hartree product hartree-product

A simple product of one-particle wavefunctions, ignoring antisymmetry. Historical predecessor to the Slater determinant.

Hartree–Fock method hartree-fock-method

A mean-field approximation in which the many-electron wavefunction is a single Slater determinant of orbitals, optimized to minimize the energy.

Heisenberg model heisenberg-model

A spin model with isotropic nearest-neighbor exchange interactions Σ S_i · S_j. Captures magnetic ordering and emerges as the strong-coupling limit of the Hubbard model.

Hohenberg–Kohn theorems hohenberg-kohn-theorems

Two theorems showing that the ground-state electron density uniquely determines the external potential (up to a constant) and that the total energy is a variational functional of the density. They are the formal foundation of DFT.

HOMO homo

The highest occupied molecular orbital. Its energy controls a system’s ionization potential and many low-energy chemical responses.

Householder QR factorization householder-qr

A factorization A = QR built by applying Householder reflectors that successively zero out subdiagonal entries. It is the standard numerically stable way to compute QR for dense matrices.

Hubbard dimer hubbard-dimer

The two-site Hubbard model with on-site repulsion U and hopping t. The smallest interacting lattice problem that exhibits FCI-like correlation effects analytically.

Hubbard model hubbard-model

A lattice model of interacting fermions with a nearest-neighbor hopping term t and an on-site Coulomb repulsion U. It is the simplest setting that captures the competition between kinetic delocalization and electronic correlation.

Huffman coding huffman-coding

A greedy algorithm that builds an optimal prefix code for a given symbol-frequency distribution by repeatedly merging the two least-frequent subtrees. Used for lossless compression.

Implicit Euler method implicit-euler-method

A backward time-stepping scheme y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}) that requires solving an algebraic equation at each step. It is unconditionally stable, making it suitable for stiff problems.

Implied volatility implied-volatility

The volatility input that, when fed into the Black–Scholes formula, reproduces the market price of an option. It is found numerically by inverting the pricing formula, typically with Newton's method on vega.

Infinite square well infinite-square-well

Synonym for the particle-in-a-box potential: zero inside an interval [0, L] and infinite outside. Often used to benchmark numerical eigensolvers.

Ionization potential ionization-potential

The minimum energy required to remove an electron from a neutral atom or molecule in its ground state. Approximated by minus the HOMO energy via Koopmans’ theorem.

Ising model ising-model

A lattice model of classical spins taking values plus or minus one with nearest-neighbor coupling. It is the prototypical example of a system with a continuous phase transition and is usually simulated by Metropolis Monte Carlo.

Itô's lemma ito-lemma

The chain rule for stochastic calculus: for f(t, X) with dX = μ dt + σ dW one has df = (f_t + μ f_x + ½ σ² f_{xx}) dt + σ f_x dW. Used to derive the Black–Scholes PDE.

Jellium jellium

A model of an interacting electron gas in a uniform positive background charge. Reference system for many DFT functionals and the homogeneous-electron-gas limit.

Kahan summation kahan-summation

A compensated-summation algorithm that accumulates the rounding error of each addition in a separate variable and adds it back on the next step. Reduces the error of summing N floating-point numbers from O(N eps) to essentially O(eps).

Kinetic energy integral kinetic-energy-integral

A one-electron matrix element T_{μν} = ⟨φ_μ| -½∇² |φ_ν⟩ contributing to the core Hamiltonian. For Gaussian primitives it has a clean closed form (Szabo–Ostlund Appendix A).

KMP algorithm kmp-algorithm

Knuth–Morris–Pratt string-matching algorithm running in O(n + m) by precomputing a longest-prefix-suffix table that lets it skip ahead on mismatch. Optimal for text-with-pattern problems with repetitive structure.

Kohn–Sham equations kohn-sham-equations

A reformulation of DFT that maps the interacting many-electron problem onto a fictitious system of non-interacting electrons sharing the same density. The single-particle equations close via an exchange–correlation potential.

Koopmans' theorem koopmans-theorem

In Hartree–Fock theory, the negative of an occupied orbital energy approximates the ionization potential, and the negative of a virtual orbital energy approximates the electron affinity. The approximation neglects orbital relaxation upon electron removal or addition.

Kruskal's algorithm kruskal-algorithm

A greedy algorithm that builds a minimum spanning tree by adding the next-cheapest edge that does not form a cycle. Uses union–find for cycle detection.

Krylov subspace krylov-subspace

The span K_m(A, v) = span{v, Av, A²v, …, A^{m−1}v}. Iterative methods like Arnoldi, Lanczos, and Davidson search for eigen- or linear-system solutions inside Krylov subspaces.

Ladder operators ladder-operators

Pairs of operators that raise and lower the eigenvalue of a chosen observable, classically illustrated by â and â^† for the harmonic oscillator. They underlie second quantization and angular-momentum algebra.

Lagrange multipliers lagrange-multipliers

Auxiliary variables introduced to enforce equality constraints during optimization. In the Hartree–Fock derivation they enforce orbital orthonormality and yield the Fock equations.

Lanczos iteration lanczos-iteration

The symmetric specialization of Arnoldi: a three-term recurrence builds an orthonormal Krylov basis and projects a Hermitian matrix onto a tridiagonal one. Standard for extremal eigenvalues of large sparse Hermitian operators.

LAPACK lapack

A standard Fortran library of dense linear algebra routines (factorizations, eigensolvers, least-squares) layered on top of BLAS. The numerical workhorse beneath NumPy, MATLAB, and quantum-chemistry codes.

Lasso regression lasso-regression

Linear regression with L1 penalty λ‖w‖_1 that produces sparse coefficient estimates by zeroing out unimportant features. Solves a convex but non-smooth optimization problem.

Legendre polynomials legendre-polynomials

The orthogonal polynomials on [-1, 1] with respect to unit weight, obtained by Gram–Schmidt on {1, x, x^2, ...}. They arise as the angular part of solutions to Laplace's equation in spherical coordinates.

Linear combination of atomic orbitals lcao

An ansatz that writes each molecular orbital as a sum of atomic basis functions weighted by coefficients to be determined variationally. It reduces solving the Hartree–Fock equations to a matrix eigenvalue problem.

Linear regression linear-regression

Fits a linear model y ≈ Xβ by minimizing ‖y − Xβ‖². The normal equations X^T X β = X^T y give a closed-form least-squares solution when X has full column rank.

Linearized augmented plane wave method lapw-method

An efficient variant of the APW method that linearizes the energy dependence of the muffin-tin radial functions, removing the need to solve a nonlinear secular equation at each k-point. The de facto standard for all-electron DFT in solids.

Log-return log-return

The continuously compounded return r_t = log(S_t / S_{t−1}). For geometric Brownian motion, log-returns are normally distributed.

LUMO lumo

The lowest unoccupied molecular orbital. Its energy controls a system’s electron affinity and serves as the first acceptor in excitations.

m-scheme m-scheme

A many-body basis built from Slater determinants of single-particle states with definite magnetic quantum number m. Standard in shell-model diagonalization and FCI implementations.

Magic numbers magic-numbers

Numbers of nucleons (2, 8, 20, 28, 50, 82, 126) at which nuclei are unusually stable, explained by closed shells in the nuclear shell model with spin–orbit coupling.

Markov chain markov-chain

A stochastic process whose next state depends only on the current state. Provides the theoretical backbone of Metropolis sampling and many MCMC methods.

Mean-field approximation mean-field

Replaces interactions among many particles with each particle’s coupling to an average (effective) field built from the others. Hartree–Fock and Kohn–Sham DFT are mean-field theories.

Memoization memoization

Caching the results of function calls keyed by arguments to avoid redundant recomputation. The top-down complement to bottom-up dynamic programming.

Merge sort merge-sort

A divide-and-conquer comparison sort that splits the array, sorts each half recursively, and merges them in O(n log n) time. Stable and well-suited to external sorting.

Metropolis algorithm metropolis-algorithm

A Markov-chain Monte Carlo method for sampling from a Boltzmann distribution: propose a local move and accept it with probability min(1, exp(-Delta E / kT)). Detailed balance guarantees the chain converges to the target distribution.

Metropolis–Hastings algorithm metropolis-hastings

A Markov-chain Monte Carlo method that generates samples from a target distribution by proposing moves and accepting them with probability min(1, π(new)/π(old)) (suitably symmetrized). Underlies the Metropolis algorithm and Ising-model simulations.

Modular exponentiation modular-exponentiation

Computes a^b mod m in O(log b) multiplications by squaring and multiplying based on the binary expansion of b. Foundation of RSA and many cryptographic primitives.

Molecular orbital molecular-orbital

A one-electron wavefunction delocalized over a molecule, typically expanded as a linear combination of atomic orbitals (LCAO). Occupied molecular orbitals build up the many-electron Slater determinant.

Møller–Plesset perturbation theory moller-plesset

Rayleigh–Schrödinger perturbation theory built on the Hartree–Fock reference, with the fluctuation potential as perturbation. MP2 is the lowest non-trivial order and a workhorse correlated method.

Monte Carlo integration monte-carlo-integration

Estimation of an integral by averaging the integrand over random samples of the domain. Error decreases as 1/sqrt(N) independent of dimension, making it the method of choice in high dimensions.

Monte Carlo method monte-carlo-method

A class of algorithms that estimate quantities by averaging over random samples, with error decaying like 1/√N independently of dimension. Underlies integration, option pricing, and statistical sampling.

Monte Carlo π monte-carlo-pi

Estimation of π by sampling random points in the unit square and counting those falling inside the inscribed quarter-circle. Pedagogical demonstration of Monte Carlo convergence.

Muffin-tin potential muffin-tin-potential

An approximation that takes the crystal potential to be spherically symmetric inside non-overlapping atomic spheres and constant in the interstitial region. Foundational for APW and KKR methods.

Multipole expansion multipole-expansion

A power-series expansion of a Coulomb-like interaction in spherical harmonics, separating angular and radial dependence. In the nuclear shell model it factors two-body interactions into Slater radial integrals times angular coefficients.

Neumaier summation neumaier-summation

A variant of Kahan summation that also corrects when the running sum becomes smaller in magnitude than the next addend. Strictly improves on Kahan in mixed-magnitude inputs.

Newton–Raphson method newton-raphson

A root-finding iteration x_{k+1} = x_k − f(x_k)/f'(x_k) that converges quadratically near a simple root. Sensitive to the initial guess and requires the derivative.

Newton's method newton-method

An iterative root finder that updates x by subtracting f(x)/f'(x). It converges quadratically near a simple root but requires a derivative and can diverge from a poor initial guess.

No-arbitrage no-arbitrage

The principle that one cannot construct a riskless portfolio with positive return in an idealized market. Underlies the existence of a risk-neutral measure and the Black–Scholes derivation.

Nuclear shell model nuclear-shell-model

A model in which nucleons fill single-particle levels of an effective central potential including a strong spin–orbit term. It explains the observed magic numbers and the systematics of low-lying nuclear states.

Numerov method numerov-method

A fourth-order finite-difference scheme tailored to second-order ODEs of the form y'' = f(x) y, with no first-derivative term. Widely used to integrate the radial Schrödinger equation.

Numerov stencil numerov-stencil

The five-point implicit difference relation underlying Numerov’s method, accurate to O(h^6) for equations of the form y'' = g(x) y + s(x). Yields a tridiagonal eigenproblem for the radial Schrödinger equation.

Operational intensity operational-intensity

The ratio of floating-point operations to bytes transferred from main memory for a kernel. The horizontal axis of the roofline diagram.

Option Greeks option-greeks

Sensitivities of an option price to underlying parameters: delta (spot), gamma (delta sensitivity), theta (time), vega (volatility), and rho (rate). Used for hedging and risk management.

Orthonormality orthonormality

A set of vectors or functions is orthonormal if it is mutually orthogonal and each element has unit norm. Required of spin orbitals in a normalized Slater determinant.

Overlap matrix overlap-matrix

The Gram matrix of basis functions, S_{μν} = ⟨φ_μ|φ_ν⟩. A non-identity overlap turns the Hartree–Fock equations into a generalized eigenvalue problem requiring symmetric orthogonalization.

Padé approximant pade-approximant

A rational function of fixed numerator and denominator degree whose Taylor expansion matches a given power series to as high an order as possible. Often dramatically extends the useful range of a divergent or slowly convergent series.

Partial differential equation pde

An equation involving partial derivatives of a multivariable unknown function. Classified into elliptic, parabolic, and hyperbolic, each with characteristic numerical strategies.

Partial pivoting partial-pivoting

A row-swap strategy in Gaussian elimination that picks the largest-magnitude entry in the current column as pivot. It controls roundoff growth and is essential for numerical stability.

Particle in a box particle-in-a-box

An infinite square-well potential with hard walls at 0 and L; eigenfunctions are sines with quantized energies E_n = n²π²ℏ²/(2mL²). The simplest non-trivial bound-state problem in quantum mechanics.

Pauli exclusion principle pauli-exclusion-principle

No two identical fermions can occupy the same quantum state simultaneously. Equivalently, the many-fermion wavefunction must change sign under exchange of any two particles.

Perturbation theory perturbation-theory

A general framework that expands eigenvalues and eigenvectors of H_0 + λV in powers of λ. In quantum chemistry it underlies Møller–Plesset and many response theories.

Phase transition phase-transition

A qualitative change in the macroscopic state of a system as a control parameter (e.g., temperature) crosses a critical value. The Ising model is the canonical lattice example.

Picard iteration picard-iteration

A fixed-point iteration that solves x = g(x) by repeatedly applying g to a starting guess. Used to prove existence of solutions to ODEs and as a practical method when g is a contraction.

Pivoting pivoting

Reordering rows or columns of a matrix during elimination to avoid small or zero pivots and to improve numerical stability. Partial and complete pivoting are the standard variants.

Plane-wave expansion plane-wave-expansion

Representation of crystal wavefunctions as Σ_G c_G e^{i(k+G)·r} over reciprocal-lattice vectors G. The native basis for periodic DFT codes and the starting point of APW/LAPW.

PMNN algorithm pmnn-algorithm

Power-Method Neural Network: parameterizes an eigenvector candidate by a neural network and trains it to minimize the residual loss until it falls below a threshold. Hybridizes power iteration with gradient-based optimization.

Power iteration power-iteration

A simple eigenvalue algorithm that repeatedly multiplies a vector by A and renormalizes; the iterate converges to the eigenvector of largest magnitude. Forms the conceptual basis of Arnoldi, Lanczos, and Davidson.

Preconditioner preconditioner

A matrix M ≈ A used to transform Av = b or Av = λv into a better-conditioned problem M^{-1}A v = M^{-1}b. The diagonal of A − μI is the standard Davidson preconditioner.

Prim's algorithm prim-algorithm

A greedy MST algorithm that grows a single tree, repeatedly adding the cheapest edge connecting the tree to a non-tree vertex. Implementations use a priority queue.

Primitive Gaussian primitive-gaussian

A single uncontracted Gaussian basis function with a specific exponent and angular factor. Multiple primitives are combined linearly with contraction coefficients to form a contracted basis function.

Put–call parity put-call-parity

The arbitrage identity C − P = S − K e^{-rT} relating European call and put prices on the same underlying and strike. Holds independently of the option pricing model.

QR factorization qr-factorization

The decomposition A = Q R with Q orthogonal (or unitary) and R upper triangular. Building block of least-squares solvers, the QR algorithm for eigenvalues, and Arnoldi.

Quick sort quick-sort

A randomized comparison sort that partitions the array around a pivot and recurses; expected O(n log n) time, in-place but not stable. Standard library default in many languages.

Rabin–Karp algorithm rabin-karp

A string-matching algorithm using a rolling hash to compare windows of text against a pattern in expected O(n + m). Naturally generalizes to multiple-pattern search.

Rayleigh plane-wave expansion rayleigh-plane-wave-expansion

Expansion of a plane wave e^{ik·r} in spherical Bessel functions and spherical harmonics. Used to match plane-wave and atomic-sphere solutions in APW and LAPW.

Rayleigh quotient rayleigh-quotient

The scalar v^T A v / v^T v, equal to an eigenvalue of A when v is the corresponding eigenvector. Stationary values of the quotient are the eigenvalues; minimization gives the lowest one.

Reciprocal lattice reciprocal-lattice

The Fourier-dual lattice of a Bravais lattice, with primitive vectors b_i satisfying a_i · b_j = 2π δ_{ij}. Hosts the Brillouin zone and indexes plane-wave momenta.

Residual residual

For an approximate eigenpair (λ, x): r = (A − λI) x. For a linear system Ax = b: r = b − Ax. Vanishing residual signals an exact solution.

Restricted Hartree–Fock restricted-hartree-fock

A closed-shell Hartree–Fock variant in which alpha and beta electrons are constrained to share the same spatial orbitals. Appropriate for systems with all electrons paired.

Rho (Greek) rho-greek

The sensitivity ∂V/∂r of an option value to the risk-free interest rate. Generally smaller in magnitude than the other Greeks for short-dated options.

Risk-neutral measure risk-neutral-measure

A probability measure under which discounted asset prices are martingales, so that option prices equal expected discounted payoffs. Equivalent to no-arbitrage in idealized markets.

RK4 rk4

The classical fourth-order Runge–Kutta method, advancing y' = f(t, y) by averaging four slope estimates per step. The default workhorse for non-stiff initial-value problems.

Roofline model roofline-model

A visual performance model that bounds attainable FLOP/s by min(peak compute, bandwidth · operational intensity). Distinguishes compute-bound from memory-bound kernels.

Roothaan–Hall equations roothaan-hall-equations

The Hartree–Fock equations expressed in a finite atomic-orbital basis as the generalized eigenvalue problem F C = S C ε. They convert the integro-differential HF problem into a matrix problem.

Runge–Kutta method runge-kutta-method

A family of single-step ODE integrators that combine several evaluations of the right-hand side per step to achieve high order of accuracy. The classical fourth-order variant (RK4) is the standard workhorse.

Schrödinger equation schrodinger-equation

The fundamental equation of non-relativistic quantum mechanics. Its time-independent form, H psi = E psi, is an eigenvalue problem for the Hamiltonian whose solutions are the stationary states and their energies.

Secant method secant-method

A derivative-free root-finder that replaces f' in Newton’s method by a finite-difference slope from the last two iterates. Convergence is superlinear (order ~1.618).

Second quantization second-quantization

A reformulation of many-body quantum mechanics in terms of creation and annihilation operators acting on Fock space. Antisymmetry of fermions is automatic via the anticommutation relations, and operators take a particle-number-independent form.

Second-quantization formalism second-quantization-formalism

A reformulation of many-body quantum mechanics in terms of creation and annihilation operators acting on Fock space, automatically enforcing fermion antisymmetry. It is the natural language of post-Hartree–Fock methods and quantum field theory.

Secular equation secular-equation

The determinantal equation det(H − E S) = 0 whose roots are eigenvalues of a generalized eigenvalue problem. Historical name for the matrix eigenproblems arising in quantum mechanics.

Self-consistent field scf

An iterative procedure in which an effective potential is built from a guess wavefunction, the resulting equations are solved for new orbitals, and the cycle repeats until the orbitals stop changing.

Sieve of Eratosthenes sieve-of-eratosthenes

An ancient algorithm that finds all primes up to N by iteratively marking multiples of each newly discovered prime. Runs in O(N log log N).

Simpson's rule simpson-rule

A Newton–Cotes quadrature rule that fits a parabola through three equally spaced points to integrate over a panel, with error O(h^5) per panel. One step up from the trapezoidal rule.

Slater determinant slater-determinant

An antisymmetrized product of single-particle spin orbitals; the simplest many-electron wavefunction that satisfies the Pauli exclusion principle.

Slater integrals slater-integrals

The radial integrals F^k and G^k that appear after a multipole expansion of 1/r_{12} between two atomic or nuclear orbitals. They store all the radial physics of two-body matrix elements.

Slater-type orbital slater-type-orbital

A one-electron basis function of the form r^(n-1) exp(-zeta r) times a spherical harmonic. Slater-type orbitals capture the correct cusp at the nucleus and the correct exponential decay at long range, but are expensive to integrate over.

Slater–Condon rules slater-condon-rules

Six matrix-element rules for one- and two-electron operators between Slater determinants that differ by zero, one, or two spin orbitals. They reduce determinantal expectation values to sums of orbital integrals.

Spatial orbital spatial-orbital

The position-dependent factor of a spin orbital, shared by an alpha- and a beta-spin electron in restricted theories. Doubly occupying a spatial orbital is the signature of closed-shell wavefunctions.

Spectral accuracy spectral-accuracy

Convergence faster than any polynomial in 1/N for smooth solutions, characteristic of well-designed spectral and pseudospectral methods. Produced by global, high-order basis expansions.

Spectral method spectral-method

A discretization that represents the solution of a differential equation as a global expansion in smooth basis functions (Fourier or Chebyshev), so that derivatives become matrix multiplications. For smooth solutions it converges exponentially in the number of basis functions.

Spectrum spectrum

The set of eigenvalues of an operator or matrix. The ground-state energy is the smallest element of the Hamiltonian’s spectrum.

Spherical harmonics spherical-harmonics

The orthonormal angular eigenfunctions Y_{lm}(θ, φ) of the squared angular-momentum operator. They give the angular factor of atomic orbitals and Slater-type orbitals.

Spin contamination spin-contamination

In unrestricted methods, the deviation of the computed total spin expectation value from its exact eigenvalue, indicating mixing of higher-spin states. Symptomatic of broken-symmetry solutions.

Spin orbital spin-orbital

A product of a spatial orbital and a one-electron spin function (alpha or beta). Slater determinants are antisymmetrized products of spin orbitals.

Spin–orbit coupling spin-orbit-coupling

A relativistic interaction between a particle’s spin and its orbital angular momentum, splitting otherwise degenerate levels. It is responsible for fine structure in atoms and shell-model magic numbers in nuclei.

Spline interpolation spline-interpolation

Piecewise-polynomial interpolation with continuity of derivatives across knots; cubic splines are the most common variety. Avoids the oscillations of high-degree polynomial interpolation.

Stable summation stable-summation

Strategies for summing many floating-point numbers while controlling roundoff growth, including pairwise, Kahan, and Neumaier summation. Crucial when adding numbers of disparate magnitudes.

Static correlation static-correlation

The portion of electron correlation arising from near-degeneracies of multiple electronic configurations, dominant in bond-breaking and biradical situations. Single-determinant methods cannot capture it.

STO-nG basis sto-ng

A minimal basis set in which each Slater-type orbital is approximated by a fixed contraction of n Gaussian primitives. STO-3G is the canonical example.

Stochastic differential equation sde

A differential equation containing a stochastic forcing term, typically a Wiener process. Models option pricing, particle diffusion, and noisy dynamical systems.

Strassen multiplication strassen-multiplication

A divide-and-conquer matrix multiplication algorithm running in O(n^{log_2 7}) ≈ O(n^{2.807}) by replacing eight subblock products with seven. First sub-cubic algorithm for matrix multiplication.

Symmetric orthogonalization symmetric-orthogonalization

The transformation X = S^{-1/2} that orthogonalizes a non-orthogonal basis while staying as close as possible to the original. Used to convert F C = S C ε into a standard eigenvalue problem.

Tensor contraction tensor-contraction

Summation over one or more shared indices of multidimensional arrays, generalizing matrix multiplication. Building the Fock matrix from the ERI tensor and density matrix is a canonical example.

Tensor-product state tensor-product-state

A many-body state expressible as a single tensor product |ψ_1⟩⊗|ψ_2⟩⊗… of subsystem states. Hartree products and unentangled wavefunctions are tensor-product states; Slater determinants are not.

Theta (Greek) theta-greek

The partial derivative ∂V/∂t of an option value with respect to time, capturing time decay. Long options typically have negative theta.

Time-independent Schrödinger equation time-independent-schrodinger-equation

The eigenvalue equation Ĥψ = Eψ for stationary states of a quantum system. Most of electronic-structure theory consists of approximate strategies for solving it.

Topological sort topological-sort

A linear ordering of the vertices of a DAG such that every edge points from earlier to later. Computed via Kahn’s algorithm or DFS post-order reversal.

Trapezoidal rule trapezoidal-rule

A quadrature rule that approximates the integral on each subinterval by the area of a trapezoid. Second-order accurate for smooth functions and exponentially accurate for periodic ones.

Trial wavefunction trial-wavefunction

A parameterized ansatz for an unknown wavefunction whose parameters are tuned to minimize the energy expectation value. It is the input to any variational calculation.

Two-electron integral two-electron-integral

A four-index integral (μν|λσ) over a product of two basis functions and the 1/r_{12} kernel, encoding all electron–electron repulsion in a finite basis. The most expensive ingredient in Hartree–Fock and post-HF methods.

Union–find union-find

Disjoint-set data structure supporting near-constant amortized union and find with path compression and union by rank. Backbone of Kruskal’s MST and connected-component algorithms.

Unrestricted Hartree–Fock unrestricted-hartree-fock

A Hartree–Fock variant that allows alpha and beta electrons to occupy distinct spatial orbitals. Useful for open-shell systems and bond-breaking, but the wavefunction is generally not a spin eigenstate.

Variance reduction variance-reduction

Techniques (control variates, importance sampling, antithetic variates, stratification) that lower the variance of a Monte Carlo estimator without changing its mean. Lets you hit a target accuracy with fewer samples.

Variational principle variational-principle

For any normalized trial wavefunction, the expectation value of the Hamiltonian is an upper bound on the true ground-state energy. Minimizing this expectation value over a parameterized family of trial wavefunctions yields the best approximation within that family.

Vega vega

The sensitivity ∂V/∂σ of an option value to volatility. Used as the derivative in Newton-Raphson root-finding for implied volatility.

Volatility smile volatility-smile

The empirical pattern of implied volatility plotted against strike, often showing higher vol for far-from-the-money options. Contradicts the constant-volatility Black–Scholes assumption.

Volatility surface volatility-surface

Implied volatility as a function of strike and time to expiration. Used to price exotic options and calibrate stochastic-volatility models.

Wavefunction wavefunction

A complex amplitude ψ whose squared modulus gives a probability density for the configuration of a quantum system. Solutions of the Schrödinger equation; central object of quantum mechanics.

Wiener process wiener-process

A continuous-time stochastic process W_t with W_0 = 0, independent Gaussian increments W_t − W_s ~ N(0, t − s), and continuous paths. Mathematical model of Brownian motion; building block of SDEs.

Zone folding zone-folding

The mapping of band-structure information from a small Brillouin zone into a larger one (or vice versa) when a unit cell is enlarged or reduced. Essential for interpreting supercell calculations.