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An atom contains an atomic nucleus and bound electrons; a neutral atom has equal numbers of protons and electrons. The atomic number specifies the number of protons, while different isotopes of the same element have different numbers of neutrons. Losing or gaining electrons produces an ion, rather than a different atomic number.

The hydrogen atom gives the simplest electronic model: one electron bound through a Coulomb interaction to one proton. Its spectral lines reveal differences between discrete energy levels. Multi-electron chemical atoms add the Pauli exclusion principle and electron correlation, so their structure is not obtained by placing every electron in the same one-electron ground state.

The neutral molecule Be-two. Its simplest filled sigma-two-s/bonding and antibonding description gives zero bond order, but mixing of accessible two-p configurations and electron correlation produce a weak covalent bond. Resolved vibrational levels establish a bound ground-state potential.^[ref]

Electron correlation concerns the dependence between electronic degrees of freedom that cannot be captured by treating each electron as moving independently. In a quantum-chemical use of the term, correlation usually means effects beyond the best single-Slater determinant Hartree-Fock method; that method already includes exchange associated with fermionic antisymmetry.

The distinction matters when comparing approximations. Dynamic electron correlation describes additional instantaneous avoidance even when one electronic configuration dominates. Static electron correlation becomes important when several configurations have comparable significance, as during some bond-dissociation processes. A correlation energy is therefore defined relative to a stated reference calculation, and the physical need for multiple configurations is different from merely allowing a single orbital to change shape.

The occupied molecular orbital of greatest energy in a specified reference electron configuration. A fixed-orbital model removes an electron from this level first; relaxation and electron correlation can modify the resulting energy.

The single-band Hubbard model balances electron hopping between localized spatial orbitals against local interaction. A common form isThe first sum is over neighbouring unordered site pairs. The parameter $t$ sets hopping, while $U_H>0$ penalizes double occupancy. At $U_H=0$ electrons occupy independent energy bands; strong repulsion suppresses local charge fluctuations and can favour local-moment behaviour. This competition was the motivation for the model's treatment of narrow-band electron correlation.^[ref]

At one electron per site and $U_H\gg |t|$, virtual double occupancy yields the antiferromagnetic scale $J_{\mathrm{ex}}=4t^2/U_H$. The two-site Hubbard dimer makes this concrete: the singlet state is lowered by $(\sqrt{U_H^2+16t^2}-U_H)/2$, while triplet states have no such shift. Its small-hopping limit gives Hubbard kinetic exchange.

Interaction conventions must match. For dimensionless spin-half operators, $-2U({\mathbf{S}}^2-3n/4)=3U{n}_{\uparrow }{n}_{\downarrow }$, so that spin-square convention has $U_H=3U$, not $U_H=U$.

Jeremy M. Merritt coauthored research on beryllium dimer.^[ref]

The neutral molecule Be-two. Its simplest filled sigma-two-s/bonding and antibonding description gives zero bond order, but mixing of accessible two-p configurations and electron correlation produce a weak covalent bond. Resolved vibrational levels establish a bound ground-state potential.

A theoretical condensed matter physicist who introduced the Hubbard model in 1963 to study electron correlation in narrow energy bands. The model balances motion of electrons between lattice sites against the Coulomb interaction cost of putting two electrons on the same site. By retaining these competing ingredients, it makes the crossover between mobile and localized electrons accessible without modeling every detail of a real solid.^[ref]

The many-body problem asks how a system of many interacting constituents behaves, given their dynamics and interactions. A simple Hamiltonian operator for $N$ nonrelativistic quantum particles contains one-particle terms and pair interactions,Even a compact expression can be difficult to solve: interactions correlate the constituents, and a tensor product of $N$ local state spaces of dimension $d$ has dimension $d^N$ before constraints or symmetries are used.

Mean-field theory replaces part of the correlated environment by an effective average, while Hartree-Fock theory uses a restricted fermionic state description. Their usefulness comes with a question: which correlations have been neglected, and do those correlations control the phenomenon of interest? Electron correlation and collective excitations show why knowing each constituent in isolation need not explain the system.

Michael C. Heaven coauthored research on beryllium dimer.^[ref]

The neutral molecule Be-two. Its simplest filled sigma-two-s/bonding and antibonding description gives zero bond order, but mixing of accessible two-p configurations and electron correlation produce a weak covalent bond. Resolved vibrational levels establish a bound ground-state potential.

The Hartree-Fock method minimizes the expectation of an interacting-electron Hamiltonian over normalized Slater determinants of orthonormal one-particle spin orbitals. It is a variational, self-consistent mean-field approximation. The determinant enforces fermionic antisymmetry exactly within that class, while each orbital responds to the mean charge density and a nonlocal exchange operator generated by the occupied orbitals.

For an electron gas, this accounts for kinetic pressure, the average electrostatic field, same-spin exchange and the associated exchange hole, and possible spin polarization. It does not account for general dynamical electron correlation, such as the correlated avoidance of opposite-spin electrons, and cannot capture all screening or collective fluctuations merely by optimizing one determinant. Uniform plane waves are an additional restriction, not the definition of Hartree-Fock itself.

In the energy functional, the one-body term is the sum of electron kinetic energy and ionic-potential energy. The direct two-body term is the Hartree energy: the classical interaction of the mean electron density with itself. Its factor $1/2$ avoids counting every pair twice. The negative Fock term comes from exchanging the two indistinguishable electron labels in the antisymmetric state. For a spin-independent interaction, spin orthogonality makes this exchange act only between equal spin components. For an orbital interacting with itself, direct and exchange contributions cancel exactly; the exchange term removes the spurious one-electron self-interaction. Ionic background terms and their prescribed cancellation must be included consistently with the direct term.

A helium atom has two electrons bound to nuclear charge $+2e$. In the nonrelativistic fixed-nucleus approximation, using atomic units,The last term couples the electrons, so a product of independent hydrogenic orbitals is not an exact solution. The dominant ground configuration is $1s^2$. Its spatial wavefunction is symmetric, so Fermi antisymmetry requires the spin singletA triplet cannot put both electrons in the same $1s$ spatial orbital. Correlation changes the exact spatial function, including its dependence on $r_{12}$, while preserving this ground-state symmetry.

A useful helium variational effective charge takes normalized orbitals ${\varphi }_{\zeta }(\mathbf{r})=({\zeta }^3/\pi {)}^{1/2}{e}^{-\zeta r}$ for both electrons and multiplies their product by ${\chi }_{00}$. The energy in Hartree units isThe reduced effective charge expresses screening. Neglecting repulsion gives $-4E_h$, while its expectation in the unscreened $\zeta =2$ product gives $-2.75E_h$; optimizing the orbital improves that trial estimate. The exact ground energy is lower than the optimized product energy, since the product omits explicit electron correlation. This calculation keeps nuclear charge two in $H$; only the trial orbital parameter is varied.

For a singly excited helium state with distinct orthonormal orbitals $a=1s$ and $b=n\ell$, the spatial combinations areThe symmetric ${\Phi }_{+}$ pairs with the antisymmetric spin singlet, while antisymmetric ${\Phi }_{-}$ pairs with the symmetric spin triplet. With the same orbitals, their leading energies areHere $J_{ab}$ is the direct Coulomb integral and $K_{ab}$ is the exchange integral. The latter is positive for the Coulomb interaction, so the triplet is lower for this fixed-orbital comparison. singlet-triplet exchange splitting in helium arises from antisymmetry and spatial correlation in the Coulomb interaction, not from inserting a separate classical spin-spin force. Spin-singlet helium states are parahelium; spin-triplet helium states are orthohelium. The two spectra approximately form separate families because the leading electric dipole operator does not change total spin.^[ref]

The lowest $1s2s$ terms are ${}^3{S}_1$ and ${}^1{S}_0$, with the triplet lower. The $1s2p$ configuration gives ${}^3{P}_{0,1,2}^{\mathrm{o}}$ and ${}^1{P}_1^{\mathrm{o}}$, where the odd parity allows singlet $P$ to connect to the even-parity ground singlet by an electric dipole transition. Both $1s2s$ levels lack an allowed direct electric dipole decay to $1s^2$: the triplet is spin forbidden, and singlet $S\to S$ is forbidden by orbital angular momentum and parity. This explains helium metastability without implying an infinite lifetime once weaker radiative processes or collisions are allowed.

The observed first $1s2s$ excitation energies are approximately $19.82$ and $20.62\mathrm{eV}$ for the triplet and singlet, respectively. Higher singly excited Rydberg series approach the He II ground-state ionization threshold; doubly excited states above that threshold can autoionize and are not all stable bound excited levels.^[ref]

Spin and spatial exchange symmetry. Write the normalized spin singlet and triplet states asThe singlet is antisymmetric under exchange and has total spin zero; the triplet is symmetric and has total spin one. For orthogonal orbitals defineIdentical spin one-half particles are fermions, so the complete state must be antisymmetric. Thus $C{\chi }_{00}$ is allowed, and $A{\chi }_{1M}$ is allowed for each triplet component. The spatially symmetric same-orbital states $|aa⟩,|bb⟩$ also require the singlet. A same-orbital triplet vanishes by the Pauli exclusion principle.

These products give a spin-adapted basis. More generally the physical state can be writtenA superposition of the singlet and triplet sectors need not factor into one spatial function times one spin function. Factorization holds for a definite-spin sector, as used in the subsequent model. If the original normalized orbitals are not orthogonal, the corresponding $ab\pm ba$ normalization is $\sqrt{2(1\pm |⟨a|b⟩{|}^2)}$; the remainder of this calculation uses the stipulated orthogonal orbitals.

Integral matrix elements and positivity. Let $a(\mathbf{r})$ and $b(\mathbf{r})$ denote the localized spatial orbitals. With fixed proton positions, the one-electron electronic operator isThe proton-proton repulsion is a separate common nuclear energy. The requested one-electron and direct two-electron elements areSince $H_1=h\otimes I$,Likewise,The integrand is nonnegative and is positive on a set of nonzero measure. This is the repulsion cost of double occupancy; the coincidence singularity is integrable for ordinary smooth localized orbitals.

One electron on each center. Suppress the normalized singlet spin factor and use $|\psi ⟩=C$. For each of the spatial products $ab$ and $ba$, the one-body sum has expectation $2E_0$. The cross terms vanish because a one-body hop leaves an orthogonal spectator orbital. The interaction contribution is the direct integral plus the exchange integral, both neglected in the model. ConsequentlyThis state has no same-center double occupancy, but remains an exchange-symmetric spatial superposition rather than assigning a distinguishable electron permanently to each proton.

Two electrons in the bonding orbital. The normalized one-electron bonding orbital is $g=(a+b)/\sqrt{2}$. The trial spatial state is $|\varphi ⟩=|gg⟩$, whose one-body expectation isExpanding the product gives $|\varphi ⟩=(|aa⟩+|ab⟩+|ba⟩+|bb⟩)/2$. The total probability of double occupancy is therefore $1/2$. Keeping only on-site Coulomb repulsion gives the intended interaction truncation,and the interaction expectation is $U/2$. HenceThis truncation must be stated. In particular, mixed Coulomb elements such as $X=⟨aa|H_{1,2}|ab⟩$ do not vanish merely by orbital orthogonality or inversion symmetry. The printed list does not explicitly name them. If they are retained and are equal and real by site symmetry, their contribution to $E_{\varphi }$ is $2X$; the requested expression uses their neglect along with the other off-site interaction terms. No numerical value of $X$ is supplied or invented.

Energy comparison and the bond. Put $x=|t|/U$ with $U>0$ and $t=-|t|$. ThenThe straight line starts at $1/2$ and crosses zero at $x=1/4$. The purely bonding-orbital trial state is better for $x>1/4$, while the singly occupied covalent trial state is better for $x<1/4$.

For localized orbitals of approximately fixed atomic size, the hopping magnitude $|t|$ decreases rapidly, typically exponentially times a polynomial, as the proton separation grows. The on-site integral $U$ remains of atomic Coulomb scale and tends to the isolated-orbital value; it does not decay as the intersite repulsion $1/R$. If the orbitals themselves are allowed to relax, $U$ also changes, so no universal monotonic law for $U(R)$ follows from the data.

At small $|t|/U$, electron correlation suppresses the ionic configurations $aa$ and $bb$, favoring a singlet with roughly one electron on each center. A single doubly occupied bonding orbital has an artificial finite ionic weight even as the centers separate. At large $|t|/U$, delocalization gains enough kinetic energy that the bonding-orbital description becomes useful and charge fluctuations are substantial.

The crossing of two restricted trial energies is not a sharp transition of the actual two-site ground state. To see this without adding empirical data, include both $C$ and the symmetric ionic state $I_{+}=(|aa⟩+|bb⟩)/\sqrt{2}$. This two-site Hubbard dimer reduces toIts ground state is $\cos \theta C+\sin \theta I_{+}$, with $\tan 2\theta =4|t|/U$ and $0\le \theta \le \pi /4$. The ionic probability isIt changes smoothly from zero to $1/2$. At large separation, the electronic lowering is $-4t^2/U$ to leading order and vanishes as hopping vanishes. An absolute molecular binding curve would additionally require $E_0(R)$, the nuclear repulsion and the neglected intersite terms; this simplified comparison alone does not determine an equilibrium bond length.

The Hartree-Fock method approximates the ground state wavefunction by a single Slater determinant, minimizing the expectation value of the Hamiltonian over its occupied orbitals. It retains fermionic exchange symmetry exactly within that determinant but omits electron correlation beyond a single Slater determinant.

Spherical averaging and self-consistent optimization impose different conditions on an atomic model. Spherical averaging replaces an angularly varying effective potential energy by one depending only on radius. It simplifies atomic orbitals into radial functions multiplied by angular functions, but does not determine those radial functions by itself.

Self-consistency instead requires the orbitals used to construct an effective potential energy to agree with the orbitals obtained by solving its Schrödinger equation. One can solve a prescribed spherical potential energy without self-consistency, or iterate a spherically averaged Hartree method model until it is self-consistent. Neither operation alone supplies all electron correlation.

Electron correlation caused by the need for multiple important configurations, often when several molecular orbitals are nearly degenerate. Stretching a homonuclear two-electron bond makes a single doubly occupied orbital retain unphysical ionic contributions; a suitable combination of configurations cancels them.

Two localized orbitals coupled by hopping, with an on-site penalty for double occupancy. At two-electron filling the competition between hopping and local repulsion can be solved exactly within the finite basis. It gives a compact example of a molecular bonding crossover and electron correlation; it is not by itself a complete real-molecule binding curve.

Vladimir E. Bondybey coauthored research on beryllium dimer.^[ref]

The neutral molecule Be-two. Its simplest filled sigma-two-s/bonding and antibonding description gives zero bond order, but mixing of accessible two-p configurations and electron correlation produce a weak covalent bond. Resolved vibrational levels establish a bound ground-state potential.