A theoretical chemist's notebook

Ab initio

Latin for “from the beginning.” Quantum chemistry built from first principles, one idea and one calculation at a time, from the spin of a single electron to the prediction of how molecules behave.

Behind this text: a hydrogen atom in an equal superposition of its 1s and 2p states, computed exactly. One beat of its electron cloud takes 0.41 femtoseconds, shown here 10¹⁶ times slower.

1929

Chemistry, from the beginning

In 1929, three years after the Schrödinger equation, Paul Dirac wrote that the laws needed for a large part of physics and “the whole of chemistry are thus completely known.” The trouble, he added, is that applying them exactly leads to equations far too complicated to solve[1].

Quantum chemistry is the century-long effort to solve those equations anyway, with approximations that are controlled, whose errors can be estimated, and that can be improved step by step toward the exact answer. A calculation is called ab initio when its only inputs are the positions and charges of the nuclei, the number of electrons, and the constants of nature. Nothing is fitted to the molecule being studied.

This page walks the road a graduate student in the field takes. It begins with the physics of a single spin, the Schrödinger equation, and the wavefunction. Then come many electrons, basis sets, and the Hartree–Fock approximation, followed by density functional theory and the ladder of correlated methods: configuration interaction, multiconfigurational and multireference methods, coupled-pair and coupled-cluster theory, perturbation theory, and finally local correlation and the many-body Green's function. The arc follows the classic graduate text by Szabo and Ostlund[2] and extends it at both ends.

  1. 1922Stern–Gerlach experiment[4]
  2. 1926Schrödinger equation[5]
  3. 1927Born–Oppenheimer separation[6]
  4. 1929Slater determinant[7]
  5. 1930Hartree–Fock[8]
  6. 1934Møller–Plesset perturbation theory[9]
  7. 1950Gaussian basis functions[10]
  8. 1951Roothaan–Hall equations[11]
  9. 1964Hohenberg–Kohn theorems[12]
  10. 1965Hedin's GW approximation[13]
  11. 1966Coupled cluster enters chemistry[14]
  12. 1980CASSCF[15]
  13. 1988Internally contracted MRCI[16]
  14. 1989CCSD(T)[17]
  15. 2013DLPNO-CCSD(T)[18, 19]

1922 · one electron, two answers

A beam of silver splits in two

In 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms between the poles of a magnet shaped so that the field was stronger near one pole than the other. A small magnet in such a field feels a force that depends on how it is oriented. Atoms pointing every which way should have smeared the beam into a continuous band. Instead it split cleanly into two[4].

The silver deposit on the collector plate was at first too thin to see. As the story goes, it appeared only when sulfur from Stern's cheap cigar turned the silver into black silver sulfide[20]. What it showed was that the magnetic moment of a silver atom, measured along the field, takes only two values. Silver has a single unpaired electron with no orbital angular momentum, so the two spots are the two possible values of that one electron's intrinsic angular momentum, its spin. The idea of spin came later, from Uhlenbeck and Goudsmit in 1925[21], and Pauli gave it a two-component wavefunction in 1927[22].

Interactive · Stern–Gerlach

atoms
0
upper spot
0
lower spot
0

Each dot is one atom. In the classical picture its magnetic moment points in a random direction and is deflected in proportion to its vertical component, so the screen fills in a continuous band. In quantum mechanics the measurement can return only spin up or spin down, each with probability ½.

Spin has no classical counterpart. It is not a small ball rotating, and it cannot be switched off. In quantum chemistry the two states are written α and β, or ↑ and ↓, and every electron in every molecule carries one of these two labels. That label will decide which electrons are allowed to share an orbital, in chapter 06.

1926–1927 · amplitudes

Up, down, and everything in between

Block the lower beam and send the upper one into a second magnet turned on its side. It splits again, half to the left and half to the right. Now keep only the left beam and send it through a third magnet, vertical like the first. The atoms that were all “up” a moment ago come out up and down in equal numbers. The sideways measurement erased what the first one established.

Quantum mechanics describes this with amplitudes. A spin state is a superposition of up and down with complex coefficients, and the probability of each outcome is the squared magnitude of its coefficient[23]:

|ψ⟩=cosθ2|↑⟩+eiϕsinθ2|↓⟩,P(↑)=cos2θ2

A spin pointing along +x is the superposition (|↑⟩ + |↓⟩)/√2. It is not a mixture in which half the atoms are secretly up and half down. It is one definite state, pointing along a different axis. Measuring along z forces one of the two outcomes, and the state becomes the one that was found. Spin along x and spin along z cannot both have definite values at once, an instance of the uncertainty principle[24].

Interactive · three magnets in a row

after magnet 2, kept
0
magnet 3 up
0
magnet 3 down
0
predicted P(up)
0.500

The first magnet keeps only spin up along z. The second, tilted by θ, keeps only spin up along its own axis, which happens with probability cos²(θ/2). The third measures along z again and finds up with probability cos²(θ/2). At θ = 90° the first measurement is completely forgotten. The sphere shows the state between the magnets (cyan) and the axis of the tilted magnet (white).

1926 · the equation of motion

How a wavefunction moves

In 1926 Erwin Schrödinger wrote down the equation that tells a quantum state how to change in time[5, 26]. For one particle of mass m in a potential V,

iℏ∂Ψ(x,t)∂t=H^Ψ(x,t),H^=−ℏ22m∂2∂x2+V(x)

The Hamiltonian Ĥ is the energy written as an operator. Its first term is the kinetic energy, which grows with how sharply the wavefunction curves. The second is the potential energy. The wavefunction is complex, and its squared magnitude |Ψ|² is a probability density[23]. Its phase is invisible in |Ψ|² but decides how the wave moves and interferes.

Interactive · a wavepacket, solved live

⟨E⟩
–
left
–
right
–
norm
–

Split-operator propagation on a 1024-point grid in atomic units (ħ = mₑ = 1), with fast Fourier transforms for the kinetic step. Color is the phase of Ψ and height is |Ψ|². The barrier height is given in units of the packet's mean kinetic energy, so values above 1 mean the particle classically cannot cross. Part of it crosses anyway, by tunneling. The shaded edges absorb the outgoing waves and count them on their side, so left plus right stays 1 because the propagation is unitary.

If the state has a definite energy E, the time dependence is only a rotating phase, Ψ(x, t) = ψ(x) e−iEt/ħ, and |Ψ|² never changes. Such stationary states solve the time-independent equation

H^ψ=Eψ

That eigenvalue problem is the one quantum chemistry solves, over and over, for the electrons of a molecule. Anything that moves, like the beating cloud at the top of this page or the packet above, is a superposition of stationary states with different energies.

1926 · the hydrogen atom

The one atom we can solve exactly

For one electron bound to one proton the time-independent equation can be solved with pen and paper. The allowed energies are Eₙ = −1/(2n²) hartree, and the wavefunctions are the orbitals of every chemistry textbook. Schrödinger's first paper on his equation did exactly this[5].

Interactive · hydrogen orbitals

energy
–
radial nodes
–
angular nodes
–
⟨r⟩
–
photon
select a level

The slice passes through the nucleus. Brightness shows |ψ|² on a compressed scale and color shows the sign of ψ. Of the seven real 4f orbitals, the four shown are those with lobes in this plane, the other three vanish on it. The level diagram uses the Rydberg constant for hydrogen, with the proton's finite mass included, and gives vacuum wavelengths. The red Balmer line, n = 3 → 2, comes out at 656.5 nm.

Two features of these orbitals matter for everything that follows. First, they have sign, and where two orbitals of the same sign overlap they add, which is the origin of chemical bonding. Second, at the nucleus the 1s orbital has a cusp, a sharp point, because the Coulomb potential diverges there[27]. Both will come back when we build molecular orbitals out of simpler functions.

The variational principle

For anything bigger than hydrogen, exact solutions are out of reach. The tool that replaces them is the variational principle. The energy of any trial wavefunction, computed with the true Hamiltonian, can never fall below the true ground-state energy:

E[ψ~]=⟨ψ~|H^|ψ~⟩⟨ψ~|ψ~⟩≥E0

So the best trial function is the one with the lowest energy, and adding flexibility can only help. Hylleraas used this in 1929 to compute the energy of helium to remarkable accuracy[28]. Try it on hydrogen with two kinds of trial function.

Interactive · minimize the energy

trial energy
–
above exact
–

Both energies are exact closed forms: E(ζ) = ζ²/2 − ζ and E(α) = 3α/2 − 2√(2α/π) hartree. The exponential reaches the exact −0.5 at ζ = 1. The single Gaussian can do no better than −4/(3π) = −0.4244 hartree, 15% too high, because it has no cusp and its tail falls off too fast.

1927 · the molecular Hamiltonian

The equation for a molecule

A molecule is a collection of nuclei and electrons that attract and repel each other through Coulomb's law. In atomic units its Hamiltonian fits on one line.

H^=−∑i12∇i2−∑A12MA∇A2−∑i,AZAriA+∑i<j1rij+∑A<BZAZBRAB

The terms are, in order, the kinetic energy of the electrons and of the nuclei, the attraction between every electron and every nucleus, and the repulsion within each pair of electrons and each pair of nuclei. Nothing else is needed for chemistry at this level, apart from relativistic corrections that matter mostly for heavy elements.

Interactive · every pair interacts

electrons
–
electron–nucleus terms
–
electron–electron terms
–
coordinates of Ψ
–

Electron positions are random snapshots, drawn near the atoms they mostly belong to. The count of electron–electron terms grows as N(N − 1)/2, and the wavefunction depends on 3N spatial coordinates plus N spins.

Born and Oppenheimer: freeze the nuclei

A proton is 1836 times heavier than an electron, so on the timescale of electronic motion the nuclei barely move. Born and Oppenheimer used this to separate the problem[6]. Fix the nuclei, solve for the electrons, and the electronic energy as a function of nuclear positions becomes the potential energy surface on which the nuclei move. The electronic Hamiltonian that the rest of this page solves is

H^el=−∑i12∇i2−∑i,AZAriA+∑i<j1rij

The last term is the whole difficulty. Without it, the electrons would move independently and the problem would split into one-electron problems, each as easy as hydrogen. With it, the motion of every electron depends on where all the others are at that instant. Every method from chapter 08 on is a strategy for living with that one term.

1925–1932 · identical particles

Electrons that refuse to share

Electrons are identical. Swap two of them and nothing measurable can change, so the wavefunction can at most change sign. For electrons it does change sign. This antisymmetry is the general form of the Pauli exclusion principle[29, 30, 31].

Give each electron a spin orbital χ(x), a spatial orbital times α or β. The simplest many-electron function, a product χ₁(x₁)χ₂(x₂)⋯, is not antisymmetric. Slater showed that a determinant is[7]:

Φ(x1,…,xN)=1N!|χ1(x1)χ2(x1)⋯χN(x1)⋮⋮χ1(xN)χ2(xN)⋯χN(xN)|

Swapping two electrons swaps two rows and flips the sign. Putting two electrons in the same spin orbital makes two columns equal and the determinant vanishes. That is the exclusion principle, built in. Antisymmetry also does something less obvious: it makes electrons of the same spin avoid each other, even with no repulsion between them.

Interactive · two electrons in a box

Two noninteracting electrons occupy two standing waves of a box. The map shows the probability density for electron 1 at x₁ (horizontal) and electron 2 at x₂ (vertical). Drag across the map to place electron 1, and the right panel shows where electron 2 can be. For two electrons of the same spin, the antisymmetric wavefunction vanishes on the whole diagonal x₁ = x₂: the Fermi hole. With opposite spins the spatial part can be symmetric and the electrons are uncorrelated.

Operators that create and destroy electrons

Determinants are clumsy to manipulate by hand. Second quantization replaces them with operators[32, 33]. The creation operator a†ₚ puts an electron into spin orbital p, the annihilation operator aₚ removes one, and antisymmetry becomes an algebraic rule: swapping two operators flips the sign.

{ap,aq†}=δpq,H^=∑pqhpqap†aq+14∑pqrs⟨pq‖rs⟩ap†aq†asar

Interactive · occupation numbers

    A determinant is a string of occupation numbers. a†ₚ or aₚ acting on it either gives zero (orbital already full or already empty) or flips that occupation and multiplies by (−1) raised to the number of occupied orbitals to the left of p. That sign is all that remains of the determinant's antisymmetry.

    In this language the matrix elements of the Hamiltonian between determinants follow simple rules worked out by Slater and Condon[7, 34]. Between two determinants that differ in more than two spin orbitals the matrix element is exactly zero, because the Hamiltonian moves at most two electrons at a time. That single fact shapes every correlated method later on this page.

    The Slater–Condon rules, for determinants built from one set of orthonormal spin orbitals.
    Determinants differ by⟨Φ|Ĥ|Φ′⟩ involves
    nothingall one-electron terms and all pairs of occupied orbitals
    one spin orbitalone element of the Fock matrix
    two spin orbitalsa single two-electron integral ⟨pq‖rs⟩
    three or moreexactly zero

    1950 · Gaussians

    Building orbitals out of Gaussians

    A computer cannot store a function. It stores numbers. So each orbital is written as a combination of fixed, known functions, the basis set, and the calculation finds the coefficients. The basis decides what a calculation can describe and how long it takes.

    The natural choice is functions shaped like hydrogen's orbitals, with a cusp at the nucleus and an exponential tail[35]. But the two-electron integrals between such functions on three or four different atoms are very hard to compute. In 1950 Boys proposed Gaussians instead[10]. The product of two Gaussians on different centers is a single Gaussian on a point between them, so every integral reduces to a few closed forms. Gaussians have the wrong shape, no cusp and a tail that dies too fast, but a few of them added together can mimic the right one.

    Interactive · fitting a Slater function with Gaussians

    overlap with 1s
    –
    H atom energy
    –
    error
    –

    Exponents and coefficients are the least-squares fits of Hehre, Stewart and Pople[36], which I reproduced for this page by maximizing the overlap with the Slater function. The energy is computed live from analytic Gaussian integrals. Watch the region near r = 0: no sum of Gaussians has a cusp.

    Modern basis sets are organized for systematic improvement. Split-valence sets such as 6-31G give the valence electrons more freedom than the core[37]. Dunning's correlation-consistent sets, cc-pVXZ with X = D, T, Q, 5, group together functions that recover similar amounts of correlation energy and add them one group at a time[38], and the augmented versions add diffuse functions for anions and weak interactions[39]. That regularity makes it possible to extrapolate to the complete basis set limit.

    Computed for this page · water, cc-pVXZ

    Water at its experimental geometry, oxygen 1s frozen in the correlated methods. The Hartree–Fock energy converges almost exponentially, with only 2.3 mEₕ between quadruple and quintuple zeta. The correlation energy converges slowly, roughly as X⁻³, and the dashed level is the two-point X⁻³ extrapolation from quadruple and quintuple zeta for CCSD(T)[40, 41].

    The slow convergence of the correlation energy has a physical cause. When two electrons meet, the exact wavefunction has a cusp in their separation r₁₂, and products of smooth orbitals cannot build one[27]. Explicitly correlated F12 methods put a function of r₁₂ into the wavefunction and reach the basis set limit with far smaller basis sets[42, 43].

    1928–1951 · the mean field

    Each electron in the average field of the others

    Hartree's idea was to replace the instantaneous repulsion between electrons by the repulsion from the average charge cloud of all the others, solve for each electron in that field, and repeat until the cloud and the orbitals agree[44]. Fock and Slater made the wavefunction an antisymmetric determinant[8, 45]. The result is the best single determinant, the Hartree–Fock approximation.

    Minimizing the energy of one determinant gives an effective one-electron equation for each orbital. The Fock operator contains the bare one-electron Hamiltonian, the Coulomb repulsion Ĵ from every occupied orbital, and an exchange term K̂ with no classical analog, which comes from antisymmetry:

    f^(1)=h^(1)+∑bocc[J^b(1)−K^b(1)],f^χa=εaχa

    Expanding the orbitals in a basis turns this into a matrix equation, derived independently by Roothaan and by Hall in 1951[11, 46]:

    F(C)C=SCε

    The Fock matrix depends on the orbitals it is used to find, so the equations are solved iteratively: guess the orbitals, build F, diagonalize, rebuild, and repeat until nothing changes. That loop is the self-consistent field (SCF). Modern programs accelerate it by extrapolating from previous iterations[47].

    Live · the SCF loop for HeH⁺

    1. Guess the density P
    2. Build F = Hcore + G(P)
    3. Solve FC = SCε
    4. New density from occupied C
    5. Converged?
    iteration
    –
    E(HF)
    –
    orbital on He
    –
    ε(occupied)
    –

    A complete Hartree–Fock program running in your browser: an STO-3G basis with the exponents of Szabo and Ostlund's worked example (ζ = 2.0925 for He, 1.24 for H), with every overlap, kinetic, nuclear-attraction and two-electron integral computed analytically. It starts from the bare-nucleus guess P = 0. At 1.4632 bohr it converges to −2.86066 hartree, the textbook value[2], and it agrees with PySCF to 10⁻¹² hartree.

    Where Hartree–Fock breaks: pulling H₂ apart

    Hartree–Fock captures more than 99% of the total energy of a molecule. The missing part, defined by Löwdin as the correlation energy[48],

    Ecorr=Eexact−EHF

    is small in relative terms but not chemically. For water the valence correlation energy is about 0.31 hartree at the basis set limit, close to 194 kcal/mol, which is comparable to the energy of both O–H bonds. And in one important situation the error is not small at all. Stretch H₂, and the restricted Hartree–Fock (RHF) determinant, with both electrons in the same bonding orbital, keeps a 50% weight of ionic configurations H⁺H⁻ even when the atoms are far apart. Letting α and β electrons have different spatial orbitals (UHF)[49] fixes the energy at large distance but breaks the spin symmetry of the true state. The exact answer needs two determinants.

    Live · H₂ in a minimal basis

    RHF
    –
    UHF
    –
    full CI
    –
    Ecorr
    –
    UHF ⟨S²⟩
    –

    STO-3G, with all integrals and all three methods computed live and checked against PySCF to 10⁻¹³ hartree. In this basis full CI is exact and needs only two determinants, σg² and σu². UHF departs from RHF at the Coulson–Fischer point[50], near 2.18 bohr in this basis, after which α and β electrons localize on opposite atoms (right panel) and ⟨S²⟩ climbs from 0 toward 1, a half-and-half mixture of singlet and triplet.

    Hartree–Fock orbital energies have a direct meaning. By Koopmans' theorem, −ε of an occupied orbital approximates the energy needed to remove that electron, if the other electrons do not respond[51]. They do respond, and they are correlated, and both effects return in chapter 16.

    1964–1965 · the density

    Everything from the density

    The wavefunction of N electrons depends on 3N coordinates. The electron density depends on three. In 1964 Hohenberg and Kohn proved that the ground-state density alone fixes the external potential, and with it everything about the ground state[12].

    So there is a functional E[ρ] whose minimum is the exact ground-state energy. Levy later gave it a constructive definition as a search over all wavefunctions with a given density[52]. Nobody knows its exact form. Early attempts in the spirit of Thomas and Fermi wrote the kinetic energy directly from the density[53], and they fail badly: they do not even bind molecules.

    Kohn and Sham's solution was to borrow the kinetic energy from a fictitious system of noninteracting electrons that has exactly the same density as the real one[54]. Those electrons occupy orbitals that obey Hartree-like equations, and everything that is hard is collected into one term, the exchange-correlation energy:

    E[ρ]=Ts[ρ]+∫v(r)ρ(r)dr+12∬ρ(r)ρ(r′)|r−r′|drdr′+Exc[ρ]

    Kohn–Sham theory is exact in principle and approximate in practice, because Exc must be modeled. Perdew organized the approximations as rungs of a ladder, each rung adding a new ingredient and, ideally, accuracy[55].

    Interactive · Jacob's ladder

    Select a rung. Each adds an ingredient to the one below. The top of the ladder is where the exact functional would sit.

    Density functional theory became the workhorse of computational chemistry because it includes correlation at roughly the cost of Hartree–Fock. Its central weakness shows up in the simplest system possible, one electron. For one electron the exchange-correlation energy must exactly cancel the electron's repulsion with itself. Approximate functionals do not quite manage, and the leftover self-interaction[66] makes them favor spreading charge out too much[67, 68]. The exact energy is a straight line between integer electron numbers[69]. Approximate functionals sag below it.

    Computed for this page · one-electron tests

    aug-cc-pVQZ basis. For one electron Hartree–Fock has no self-interaction and is exact in the basis, so it is the reference. Left: a hydrogen atom holding N electrons, 0 ≤ N ≤ 1, using fractional occupation. The functionals bow below the straight line by up to 33 kcal/mol at N = ½. Right: the H₂⁺ bond stretched to 8 Å. The exact energy goes to that of a hydrogen atom, but the functionals keep the electron spread over both nuclei and end 56 kcal/mol (PBE) too low.

    Hybrid functionals mix in a fraction of exact Hartree–Fock exchange[63, 64], which reduces this error but does not remove it. Dispersion, which semilocal functionals miss entirely, is usually added as a separate correction[70]. Hundreds of functionals now exist, and benchmarking them is a field of its own[71]. Walter Kohn and John Pople shared the 1998 Nobel Prize in Chemistry, one for the density and one for systematic wavefunction methods[72, 73]. The rest of this page follows Pople's road, where every approximation can be improved step by step toward the exact answer.

    1950s · configuration interaction

    Mixing in excited determinants

    The Hartree–Fock determinant is one term. In a given basis, the exact wavefunction is a superposition of every determinant you can build by moving electrons from occupied to virtual orbitals. Diagonalize the Hamiltonian in that space and you have full configuration interaction, the exact answer for that basis[10, 74].

    |Ψ⟩=c0|Φ0⟩+∑iacia|Φia⟩+∑i<j,a<bcijab|Φijab⟩+∑i<j<k,a<b<ccijkabc|Φijkabc⟩+⋯

    Here i, j, k label occupied spin orbitals and a, b, c virtual ones. The catch is the number of determinants. Distributing N electrons among K orbitals grows combinatorially, and full CI is possible only for small molecules in small basis sets. Calculations with about a trillion determinants have been run on supercomputers[75].

    Interactive · how big is full CI?

    –

    Determinants with equal numbers of α and β electrons: C(K, N/2)². The bar is logarithmic, from 1 to 10⁸⁰, about the number of atoms in the observable universe. Storing one coefficient per determinant in double precision takes 8 bytes. The water numbers count all electrons. Freezing the oxygen 1s, as in the calculations on this page, shrinks them somewhat.

    Most of that matrix is empty. With Hartree–Fock orbitals, singly excited determinants do not mix directly with the reference, a result known as Brillouin's theorem, and by the Slater–Condon rules the reference couples to nothing beyond double excitations. The correlation that matters enters first through doubles.

    Computed for this page · the full CI matrix of water

    Hover over the matrix to read an element.

    Left: water in the STO-3G basis with the oxygen 1s frozen, the 65 determinants of the right symmetry, grouped by excitation level from the Hartree–Fock determinant (0 to 8). Brightness is log₁₀|HIJ|, black is exactly zero. The empty strips in the first row and column are Brillouin's theorem and the Slater–Condon rules. Right: how the exact (full CI) wavefunction of water in the 6-31G basis divides its weight among excitation levels, at the equilibrium bond length and with both O–H bonds stretched.

    Near equilibrium the exact wavefunction is 95.5% Hartree–Fock and 4.3% doubles, with quadruples next and singles and triples tiny. So a natural approximation is CI with singles and doubles, CISD. It is variational and affordable, but it has a serious flaw. It is not size-consistent: the energy of two molecules far apart is not the sum of their separate energies[76]. The reason is that two molecules each doing a double excitation make a quadruple excitation of the pair, and CISD does not have it.

    Live · N copies of H₂, far apart

    Minimal-basis H₂ molecules with no interactions between them, integrals computed live. Full CI gives the same correlation energy per molecule however many there are. CISD gives Ecorr = Δ − (Δ² + N K²)½ for N molecules, so the energy per molecule shrinks toward zero as N grows. Coupled-cluster doubles, chapter 13, is exact here for every N.

    Large CI expansions are solved iteratively. Davidson's method finds the lowest eigenvectors without ever storing the matrix[77], and determinant-based algorithms made full CI routine for small problems[78, 79]. Modern descendants select only the important determinants[80, 81], sample them stochastically[82], or compress the wavefunction as a matrix product state with the density matrix renormalization group[83, 84].

    1939–1985 · active spaces

    When one determinant is not enough

    Pull apart the two atoms of N₂ and its triple bond must become three unpaired electrons on each nitrogen. Describing that takes many determinants with large weights, and no single determinant is even qualitatively right. This is static, or nondynamic, correlation.

    The remedy is to choose a small set of active orbitals, those that change occupation as the bond breaks, do full CI inside them, and optimize the orbitals and the CI coefficients together. That is the complete active space SCF method, CASSCF[15, 85]. The idea of mixing configurations self-consistently goes back to Hartree, Hartree, and Swirles in 1939[86], and robust second-order algorithms made it practical[87].

    The diagnostic is the occupation of the natural orbitals, the eigenvectors of the one-electron density matrix[88]. In a single-determinant world they are exactly 2 or 0. Values far from 2 and 0 mean static correlation.

    Computed for this page · N₂, CASSCF(6,6)

    6-31G basis, six electrons in the six orbitals made from the nitrogen 2p orbitals: σg, two πu, two πg*, σu*. At equilibrium the bonding orbitals hold nearly 2 electrons. At 3 Å all six approach 1, three broken bonds.

    CASSCF gets the shape of the curve right but not its depth. It recovers static correlation inside the active space and almost none of the dynamic correlation outside it. For N₂ in this basis it lies about 75 to 94 mEₕ above the exact energy all the way out. The error is large but nearly constant, which is exactly what static correlation alone should give. Dynamic correlation is then added on top, by perturbation theory (CASPT2[89], NEVPT2[90]) or by multireference CI.

    1974–1988 · many references

    Excitations from every reference

    Take every determinant in the CASSCF wavefunction as a reference, and add all single and double excitations out of each one. That is multireference CI, MRCISD, and for bond breaking it remains one of the most accurate tools available[92, 16, 93].

    Below is the classic test, breaking the triple bond of N₂, against the exact full CI curve in the same basis[94]. Each single-reference method starts from the RHF determinant, which becomes worse and worse as the bond stretches. Switch methods on and off, and look at the error panel.

    Computed for this page · breaking N₂

    6-31G basis with both nitrogen 1s orbitals frozen, so every method works in the same space of 16 orbitals and 10 electrons, where full CI is exact. MRCISD here is uncontracted: every configuration within a double excitation of the CASSCF(6,6) configurations, with all of its determinants so the space is spin-complete, and the singlet found by diagonalizing in that subspace of the full CI space. CASSCF and MRCISD were both constrained to the singlet, which matters near 3 Å where a quintet state lies within about a millihartree. +Q is the Davidson correction[95]. Points where a method failed to converge are left out.

    The pattern is the one that has made multireference methods indispensable. CCSD(T), superb near equilibrium with an error of 1.8 mEₕ at 1.10 Å, falls below the exact curve as the bond stretches, by 14.3 mEₕ at 1.8 Å, an unphysical dip. MP2 goes wrong even faster. MRCISD stays between 1.9 and 2.4 mEₕ above full CI across the whole curve, so its errors largely cancel in energy differences. Its remaining flaw is that, like any truncated CI, it is not size-consistent. The Davidson correction estimates the missing quadruples and brings it within about 1.4 mEₕ of full CI. Fittingly, Langhoff and Davidson introduced that correction in a paper on N₂[95].

    Uncontracted MRCI grows quickly with the size of the active space. Internally contracted MRCI, which excites the CASSCF wavefunction as a whole rather than each determinant separately, made the method practical for real molecules[16].

    1958–1989 · electron pairs

    Pairs, coupled pairs, and the exponential

    Doubles carry most of the correlation, so correlation is, to a first approximation, a property of electron pairs[96, 97]. The pair theories of the 1960s and 1970s took that literally, and coupled-cluster theory made it exact in form.

    The simplest version, the independent electron pair approximation (IEPA), correlates each pair of occupied spin orbitals as if the others were frozen and adds up the pair energies. It overshoots, because pairs that share an electron each claim part of the same correlation. For water in the 6-31G basis it gives 128% of the exact correlation energy. Coupled electron pair approximations (CEPA) let the pairs feel each other through shifts in their equations[98, 99].

    The decisive step came from nuclear physics. Coester and Kümmel wrote the wavefunction as an exponential[100, 101], and Čížek brought the idea into quantum chemistry[14, 102]:

    |Ψ⟩=eT^|Φ0⟩,eT^=1+T^+12!T^2+13!T^3+⋯,T^=T^1+T^2+⋯

    Truncate T at doubles and the exponential still contains ½T₂², simultaneous pairs of independent double excitations, which are quadruples. They come with no new parameters, and they are exactly what CISD was missing. That is why coupled cluster is size-extensive, and the plot below shows that nature agrees. The quadruple-excitation coefficients of the exact wavefunction come out close to products of double-excitation coefficients.

    Computed for this page · why the exponential works

    Water, 6-31G, oxygen 1s frozen, symmetric O–H stretch. Each dot is one of the 600 largest quadruply excited determinants of the exact full CI wavefunction (intermediate normalization). The horizontal axis is the prediction ½C₂² built only from the exact double-excitation coefficients, applied with the true operator algebra including signs. Points on the diagonal mean the quadruples are disconnected products of pairs. At equilibrium the median deviation among the 50 largest is 9%.

    Practical coupled-cluster doubles (CCD) programs appeared in 1978[103, 104], singles and doubles (CCSD) in 1982[105], and the perturbative triples correction (T) in 1989[17]. CCSD(T) became the “gold standard” of quantum chemistry. For well-behaved molecules near equilibrium it typically lands within a millihartree or two of the exact answer in the same basis, at a cost that grows as the seventh power of system size. Full triples (CCSDT) and quadruples (CCSDTQ) close the remaining gap[106, 107]. Reviews by Bartlett and Musiał[108] and by Crawford and Schaefer[109] are good entry points.

    Computed for this page · the whole hierarchy, against the exact answer

    Water, 6-31G, oxygen 1s frozen. Bars show E(method) − E(full CI) in mEₕ. Positive means too high. Negative means below the exact energy, which no variational method can do. CEPA(0) is the linearized CCD. IEPA, CEPA(0) and CID/CISD were computed by restricting the exact full CI problem to the corresponding determinants, and CC energies come from PySCF. Bars beyond ±60 mEₕ are cut and labeled.

    Two lessons stand out. Near equilibrium the coupled-cluster ladder converges steadily, from 1.51 mEₕ for CCSD to 0.52 for CCSD(T), 0.44 for CCSDT and 0.01 for CCSDTQ. With both bonds stretched to twice their length, the reference determinant carries only 63% of the wavefunction, and CCSD(T) falls 4.8 mEₕ below the exact energy. The (T) correction assumes the reference is good. When it is not, the multireference methods of chapters 11 and 12 take over.

    1934 · perturbation theory

    Correlation as a perturbation

    Split the Hamiltonian into a sum of Fock operators, whose ground state is the Hartree–Fock determinant, plus everything left over. Then expand the energy in powers of the leftover[9]. Møller–Plesset perturbation theory reproduces Hartree–Fock at first order, and the first correction appears at second order:

    E(2)=14∑ijocc∑abvirt|⟨ij‖ab⟩|2εi+εj−εa−εb

    MP2 is the cheapest correlated method, with cost growing as the fifth power of system size, and for water near equilibrium it captures 94% of the exact correlation energy in the basis used here. Third order adds pair–pair couplings, and fourth order is the first to see triples and quadruples[110, 111].

    ia jb
    The second-order Goldstone diagram. Dashed lines are interactions, upward lines are electrons promoted into virtual orbitals, downward lines are the holes they leave. Only linked diagrams like this one contribute to the energy, which makes every order size-extensive[112, 113].

    Computed · what fourth order sees

    E⁽⁴⁾ of water (6-31G) split by excitation level of the second-order wavefunction, in mEₕ. The renormalization term is included with the quadruples, as usual. Triples are already larger than singles, which is why a triples correction belongs in coupled cluster.

    Does the series converge? Because MPn energies to any order can be computed from the full CI machinery, this was tested directly in the 1980s[114, 115]. The answer is: not always. With stretched bonds the series oscillates, and with diffuse basis functions it can diverge even for simple molecules near equilibrium[116, 117]. Here it is for water, to fortieth order.

    Computed for this page · the Møller–Plesset series to 40th order

    E(MPn) − E(FCI)
    –
    sign
    –

    Water, 6-31G, oxygen 1s frozen. Each order was computed with the Rayleigh–Schrödinger recursion in the full CI space, using H₀ = sum of Fock operators. The vertical axis is |E(MPn) − E(full CI)| on a log scale. Filled dots lie above the exact energy and open dots below. Errors below 1 nanohartree are drawn on the floor, the numerical precision of the calculation. The MP2 energy from this recursion agrees with PySCF's MP2 to 10⁻¹³ hartree.

    In practice MP2 lives on as a fast first estimate and as an ingredient of double-hybrid functionals and of explicitly correlated and local methods, while higher orders mostly survive inside coupled cluster. CCSD is exact through third order and contains the singles, doubles and quadruples parts of fourth order, and (T) adds a perturbative estimate of the triples[17, 118].

    1983–today · locality

    Correlation is nearsighted

    Canonical CCSD(T) costs grow as the seventh power of system size. Double the molecule and the calculation takes 128 times longer. But dynamic correlation is short-ranged. Electrons far apart correlate only through dispersion, which falls off as R⁻⁶[119]. Pulay saw that this could be exploited[120, 121].

    Canonical Hartree–Fock orbitals spread over the whole molecule. Rotating the occupied ones among themselves changes nothing physical, and a suitable rotation makes them compact bond and lone-pair orbitals[122, 123]. In that basis the correlation energy splits into pair contributions, and their decay with distance is the nearsightedness of electronic matter made visible[124, 125].

    Computed for this page · MP2 pair energies in decane

    pairs kept
    –
    correlation energy kept
    –
    correlation energy lost
    –

    All-trans n-decane, cc-pVDZ, carbon 1s frozen, 31 Boys-localized valence orbitals. The MP2 first-order wavefunction does not depend on how the occupied orbitals are rotated, so the canonical amplitudes were transformed to the localized basis and the 496 pair energies sum to the canonical MP2 correlation energy, -1.4491 hartree, to 10⁻¹⁵. The guide line falls as R⁻⁶. Select an orbital center above to see its partners.

    Modern local methods build on this in three ways. They screen pairs by distance or by a cheap estimate of their energy. For each surviving pair they compress the virtual space into a few pair natural orbitals, an idea from the 1960s and 1970s[126, 98] revived in 2009[127]. And they organize everything with sparse data structures so cost grows nearly linearly with size[128, 18, 19, 129]. Domain-based local pair natural orbital CCSD(T), local natural orbital CCSD(T), and explicitly correlated local methods now reach molecules with hundreds of atoms close to canonical accuracy[130, 131]. A complementary route cuts the system into fragments and adds up their energies in a many-body expansion, which is how water clusters are often analyzed[132, 133].

    Interactive · what scaling means

    Formal scaling of the cost with system size N, relative to the starting system. Local methods approach linear scaling once the molecule is larger than the range of the correlation, so their real prefactors matter more than the exponent shown.

    1949–today · propagators

    Adding or removing one electron

    Photoelectron spectroscopy knocks one electron out of a molecule and measures what that costs. The one-particle Green's function describes exactly that process: an electron or a hole propagating through the interacting system. Its poles are the ionization energies and electron affinities, and its residues, the pole strengths, set how intense each line is.

    The interacting Green's function G follows from the Hartree–Fock one, G₀, through the Dyson equation[134], with all many-body effects packed into the self-energy Σ:

    G(ω)=G0(ω)+G0(ω)Σ(ω)G(ω),ω=εp+Σpp(ω)

    The second equation, in the diagonal approximation, says that an ionization energy is the Koopmans value corrected by a frequency-dependent self-energy. At second order the self-energy is a sum of simple poles, one for each way the system can rearrange around the hole, and the equation can be solved graphically.

    Computed for this page · Dyson's equation for water

    Koopmans
    –
    second order
    –
    shift
    –

    Water, cc-pVDZ, from the Hartree–Fock orbitals and integrals. The straight line is ω, the curve is εp + Σpp(ω) with Σ at second order, and the ticks along the bottom mark its poles. The quasiparticle solution is the crossing nearest the Koopmans value. For the inner-valence 2a₁ orbital the poles crowd in and the single-crossing picture starts to fail.

    Vertical ionization energies of water in eV, aug-cc-pVTZ basis, computed for this page. Experimental values are from photoelectron spectra[135, 136].
    OrbitalKoopmansGF2G₀W₀@PBEEOM-IP-CCSDExperiment
    1b₁13.8911.5611.6412.6112.62
    3a₁15.9213.8413.9114.83≈ 14.7
    1b₂19.5318.1818.2619.01≈ 18.5
    2a₁36.8331.9529.6932.80≈ 32

    Koopmans' theorem overestimates the first ionization energy by more than a volt, because it ignores how the remaining electrons relax around the hole and how correlation changes. Diagonal second order overcorrects. Hedin's GW approximation replaces the bare interaction with a screened one[13]. It became the standard for band gaps of solids[137] and is now benchmarked on molecules[138, 139]. Equation-of-motion coupled cluster brings the coupled-cluster hierarchy to ionized states[140, 141]. Here it reproduces the first ionization energy of water to within 0.01 eV. The algebraic diagrammatic construction organizes the self-energy order by order[142].

    Deep in the valence shell the quasiparticle picture itself breaks down. The 2a₁ hole of water relaxes strongly[136]. At second order its main line keeps a pole strength of only 0.54, with 24 other solutions within 10 eV of it, and in many molecules inner-valence ionization intensity splinters into many satellite lines, a breakdown of the molecular orbital picture first explained with Green's functions[143, 144]. Recent work pushes Green's function methods toward strong correlation and large systems, with self-consistent second order[145], auxiliary second-order expansions[146], and low-scaling GW for thousands of atoms[147]. The locality of chapter 15 reappears here too.

    coda · two axes

    The map

    Pople summarized the field as a chart with two axes: how completely the electrons are correlated, and how complete the basis set is. The exact solution of the Schrödinger equation sits in the far corner where both are complete[73].

    Computed for this page · water on Pople's chart

    Water at its experimental geometry, oxygen 1s frozen. Each cell gives the energy above the best estimate in mEₕ, where the best estimate is the CCSD(T) correlation energy extrapolated from quadruple and quintuple zeta (X⁻³) added to the quintuple zeta Hartree–Fock energy, and the CBS column extrapolates each method the same way. Brighter is closer. At the same geometry Helgaker and coworkers report the same Hartree–Fock energies to the microhartree[40]. Their correlated energies are lower than these because they also correlate the oxygen 1s electrons.

    Every chapter of this page is a step along one of those two axes. The ideas are a century old and still moving: the same exponential, the same pairs, the same propagators, made faster and more local each decade. For a deeper path through the same road, the books by Szabo and Ostlund[2], Helgaker, Jørgensen, and Olsen[148], and Shavitt and Bartlett[149] are where to go next.

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