Molecular orbital theory for homonuclear diatomics

Intuition [Beginner]

Two hydrogen atoms, brought close enough together, form an H2 molecule. The bond is so stable that pulling the atoms apart again costs about 436 kilojoules per mole of bonds broken. Two helium atoms, brought just as close, refuse to bond — He2 simply does not exist as a stable molecule under ordinary conditions. Why is the first stable and the second not?

The Lewis picture you may have seen first says: H has one electron each, the two pair up to make a shared pair, and a shared pair is what we call a bond. He has two electrons each; there is no room to share because everyone is already paired. This picture predicts the outcome correctly, but it does not explain why sharing produces stability or why a full shell is so unfriendly to bond formation.

Molecular-orbital theory gives the deeper answer. When two atoms come together, the electrons no longer belong to one atom or the other. They occupy new orbitals that stretch across both atoms — molecular orbitals. Each molecular orbital has a definite energy, and electrons fill these orbitals from the bottom up, two per orbital, just like in an atom.

The key fact is that two atomic orbitals always combine to give two molecular orbitals: one with energy lower than the original atomic orbitals, and one with energy higher. The lower one is a bonding orbital — putting an electron there releases energy and stabilises the molecule. The higher one is an antibonding orbital — putting an electron there destabilises the molecule, by exactly the opposite amount (slightly more, in fact). The visual you should hold in your head: the bonding orbital has the two atomic orbitals adding constructively, with electron density piling up between the nuclei; the antibonding orbital has them adding destructively, with a node between the nuclei.

So for H2: each atom brings one electron, and there are two new molecular orbitals to fill. Both electrons drop into the lower bonding orbital. The molecule is more stable than the two separated atoms, and a bond exists. For He2: each atom brings two electrons, so four electrons total. Two go into the bonding orbital, but the other two are forced up into the antibonding orbital. The energy gained from the first pair is roughly cancelled (a little less, in fact) by the energy spent on the second pair. The net is no benefit, so He2 does not form.

A useful summary number is the bond order:

$$\text{bond order}=\frac{1}{2}(\text{electrons in bonding orbitals}-\text{electrons in antibonding orbitals}).$$

H2 has bond order 1: a single bond. He2 has bond order 0: no bond. N2 turns out to have bond order 3: a triple bond, and it is famously hard to react — the triple bond holds it together. O2 has bond order 2, a double bond, but with a famous twist taken up below.

The most striking prediction of MO theory is about oxygen. The Lewis structure of O2 draws a double bond with all four non-bonding electrons paired off; it predicts that O2 is diamagnetic (not attracted to a magnet). Liquid oxygen is in fact paramagnetic — pour it between the poles of a magnet and it sticks.

MO theory explains this. The MO diagram for O2 places its top two electrons in two different antibonding orbitals of equal energy, and by Hund's rule these two electrons remain unpaired with parallel spins. Two unpaired electrons give O2 a permanent magnetic moment. The Lewis picture draws something qualitatively wrong about the molecule's magnetism; MO theory gets it right.

This is the headline of the unit: classical "shared electron pair" thinking gets you the right bond order most of the time, but for O2 it fails on magnetism, and for the rest of the second-row diatomics it cannot explain the systematic patterns in bond lengths, bond strengths, and ionisation energies that MO theory reproduces from first principles. The price you pay for MO theory is letting go of the Lewis-bond-as-localised-pair picture and accepting that bonding is a property of the molecule as a whole.

Visual [Beginner]

Picture two 1s orbitals on adjacent hydrogen atoms — two spheres of electron density centred on the two nuclei. Bring them close. Two new things appear in the picture.

Adding the orbitals (with the same sign on both sides) builds an amplitude that is large between the nuclei: the bonding orbital, called ${\sigma }_g$ or "sigma bonding." Electrons in this orbital sit in the region between the two nuclei and feel attraction from both at once — that is what holds the molecule together. Subtracting the orbitals (with opposite signs on the two sides) gives an amplitude that is zero on the midplane and grows in magnitude outside the bond axis on each side: the antibonding orbital, called ${\sigma }_u^{\ast }$ or "sigma antibonding." Electrons here sit outside the internuclear region and feel less attraction than they did in the isolated atoms.

For H2, both electrons go into ${\sigma }_g$. For He2, two go into ${\sigma }_g$ and two are forced into ${\sigma }_u^{\ast }$; the antibonding pair more than wipes out the bonding pair, and the molecule doesn't form.

For the second-row diatomics (Li2 through F2), the picture grows. Each atom now contributes a 2s orbital and three 2p orbitals (px, py, pz, with z along the bond axis). Combining gives eight molecular orbitals total, arranged in order of energy.

The two 2s orbitals make a bonding/antibonding pair. The two pz orbitals (along the bond axis) make a sigma bonding/antibonding pair. The four perpendicular orbitals (px and py) make two pairs of "pi" orbitals — bonding and antibonding — with the special feature that the pi bonding pair are degenerate (same energy) and so are the pi antibonding pair. Filling these two electrons at a time, with Hund's rule for degenerate orbitals, gives the standard second-row MO diagram.

Worked example [Beginner]

Take O2. Each oxygen atom has 8 electrons; an O2 molecule has 16 in total. Two are in the deep $1s$ orbitals on each oxygen (4 total — chemically inert "core" electrons). That leaves 12 electrons to fill the bonding-relevant orbitals built from the 2s and 2p atomic orbitals.

Step 1. List the MOs for O2 in order of increasing energy:

$${\sigma }_g(2s),{\sigma }_u^{\ast }(2s),{\sigma }_g(2p_z),{\pi }_u(2p_x),{\pi }_u(2p_y),{\pi }_g^{\ast }(2p_x),{\pi }_g^{\ast }(2p_y),{\sigma }_u^{\ast }(2p_z).$$

(There is a subtlety about the ordering of ${\sigma }_g(2p_z)$ relative to the ${\pi }_u(2p)$ pair that switches between N2 and O2; for O2 the ordering above is right.)

Step 2. Fill these orbitals with the 12 valence electrons, two per orbital, lowest first:

Orbital	Type	Electrons
${\sigma }_g(2s)$	bonding	2
${\sigma }_u^{\ast }(2s)$	antibonding	2
${\sigma }_g(2p_z)$	bonding	2
${\pi }_u(2p_x),{\pi }_u(2p_y)$	bonding pair	4
${\pi }_g^{\ast }(2p_x),{\pi }_g^{\ast }(2p_y)$	antibonding pair	2

The last two electrons land in the degenerate ${\pi }_g^{\ast }$ pair. By Hund's rule they occupy different spatial orbitals and have parallel spins. So O2 has two unpaired electrons.

Step 3. Count bonding and antibonding electrons:

• Bonding electrons: $2+2+4=8$.
• Antibonding electrons: $2+2=4$.

Bond order $=\frac{1}{2}(8-4)=2$. A double bond.

Step 4. Predict properties:

• Bond order 2 — agrees with the experimental bond dissociation energy ($\approx 498$ kJ/mol) and bond length ($\approx 121$ pm), between an O-O single bond (peroxide $\approx 145$ pm) and a true O=O fixed double bond.
• Two unpaired electrons — predicts paramagnetism. Confirmed experimentally: liquid O2 sticks to a magnet.

What this tells us: MO theory gets bond order and magnetism together from one filling diagram. The Lewis structure $\mathrm{O}=\mathrm{O}$ gets bond order 2 but predicts zero unpaired electrons and therefore diamagnetism — wrong for O2. The structural origin of the discrepancy is that the highest two electrons sit in degenerate antibonding orbitals that Lewis cannot draw.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Consider two atoms $A$ and $B$ with atomic orbitals ${\varphi }_1^A,\dots ,{\varphi }_n^A$ on atom $A$ and ${\varphi }_1^B,\dots ,{\varphi }_m^B$ on atom $B$. The linear-combination-of-atomic-orbitals (LCAO) ansatz writes each molecular orbital $\psi$ as a linear superposition of all the atomic orbitals:

$$\psi =\sum _{\mu =1}^n{c}_{\mu }^A{\varphi }_{\mu }^A+\sum _{\nu =1}^m{c}_{\nu }^B{\varphi }_{\nu }^B=\sum _{i=1}^{n+m}{c}_i{\varphi }_i,$$

where in the second form we re-index $\{{\varphi }_{\mu }^A\}\cup \{{\varphi }_{\nu }^B\}$ as $\{{\varphi }_i{\}}_{i=1}^{n+m}$ and similarly for the coefficients $c_i$. The atomic-orbital set $\{{\varphi }_i\}$ is called the basis set. The MO is a vector in the finite-dimensional subspace 01.01.03 of $L^2({\mathbb{R}}^3)$ spanned by the chosen basis.

The variational principle asserts that the molecular-electronic energy $E$ associated with the trial function $\psi$, namely

$$E[\psi ]=\frac{⟨\psi |\hat{H}|\psi ⟩}{⟨\psi |\psi ⟩},$$

satisfies $E[\psi ]\ge E_0$, the true ground-state energy of the (one-electron) Hamiltonian $\hat{H}$. Equality holds when $\psi$ is the true ground state. Minimising $E[\psi ]$ over the LCAO coefficients gives the best approximation to the true MO that the chosen basis can produce.

Setting $\partial E/\partial c_i^{\ast }=0$ for each $i$ yields the generalised eigenvalue problem (the Roothaan-Hall equations in the multi-electron case):

$$\sum _j{H}_{ij}{c}_j=E\sum _j{S}_{ij}{c}_j,$$

where the matrix elements are

$$H_{ij}:=⟨{\varphi }_i|\hat{H}|{\varphi }_j⟩,S_{ij}:=⟨{\varphi }_i|{\varphi }_j⟩.$$

$H_{ij}$ is called the Hamiltonian matrix (or Fock matrix in the multi-electron context); $S_{ij}$ is the overlap matrix. Both are Hermitian; $S$ is positive-definite (since $\{{\varphi }_i\}$ is linearly independent). Non-zero coefficient vectors $(c_1,\dots ,c_{n+m}{)}^{\top }$ exist exactly at energies $E$ satisfying the secular equation

$$\det (H_{ij}-E{S}_{ij})=0,$$

a polynomial of degree $n+m$ in $E$. Its $n+m$ roots are the molecular-orbital energies; the eigenvectors give the LCAO coefficients 01.01.08.

Symmetry labels. For a homonuclear diatomic $A_2$ the relevant point group is $D_{\infty h}$. The irreducible representations split each MO according to its behaviour under the cylindrical-symmetry operations and under inversion through the bond midpoint. The labels used in this unit:

• $\sigma$ ($\Sigma$-type) — MOs with no angular nodes around the bond axis ($m=0$).
• $\pi$ ($\Pi$-type) — MOs with one angular node ($|m|=1$); always come in degenerate pairs.
• $g$ (gerade) — even under inversion through the midpoint.
• $u$ (ungerade) — odd under inversion.
• $\ast$ — antibonding (one or more nodes between the nuclei).

So ${\sigma }_g$ is "sigma, even, bonding," ${\sigma }_u^{\ast }$ is "sigma, odd, antibonding," etc. The character-theory underpinning 07.01.03 makes these labels track the irreducible representations of $D_{\infty h}$ on the MO space.

Minimal-basis H2 ($1s$ basis). Take ${\varphi }_A=1s_A$, ${\varphi }_B=1s_B$ — one orbital per atom. By the symmetry $A\leftrightarrow B$ the Hamiltonian matrix has diagonal $H_{AA}=H_{BB}=:\alpha$ and off-diagonal $H_{AB}=H_{BA}=:\beta$. The overlap matrix has diagonal $S_{AA}=S_{BB}=1$ (normalised atomic orbitals) and off-diagonal $S_{AB}=S_{BA}=:S$. The secular equation reads

$$\left|\begin{array}{cc}\alpha -E& \beta -ES\\ \beta -ES& \alpha -E\end{array}\right|=0,$$

giving $(\alpha -E{)}^2=(\beta -ES{)}^2$. The two roots are

$$E_{\pm }=\frac{\alpha \pm \beta }{1\pm S}.$$

For the standard sign convention $\alpha <0$ (atomic orbital binding energy), $\beta <0$ (resonance integral, bonding), and $0<S<1$ (positive overlap), $E_{-}$ — corresponding to the $+$ in numerator and denominator combined as $(\alpha +\beta )/(1+S)$ — is the lower (bonding) energy: this is ${\sigma }_g$. $E_{+}$ is the upper (antibonding) energy: ${\sigma }_u^{\ast }$. The corresponding unnormalised coefficients are

$${\psi }_{{\sigma }_g}\propto {\varphi }_A+{\varphi }_B,{\psi }_{{\sigma }_u^{\ast }}\propto {\varphi }_A-{\varphi }_B.$$

The bonding orbital adds the two AOs in phase (electron density piled up between the nuclei); the antibonding orbital subtracts them (a node at the midplane).

Aufbau filling and Hund's rule. With the orbital energies in hand, electrons are placed in MOs in order of increasing energy, two per spin-orbital (Pauli exclusion 12.01.02 pending). Where degenerate orbitals are partially filled, Hund's first rule says the configuration with maximum total spin is the ground state — the spins remain unpaired and parallel until each degenerate orbital has one electron.

Counterexamples to common slips

• The number of MOs equals the number of AOs in the basis, not the number of bonds. Two AOs give two MOs (one bonding, one antibonding), regardless of how many electrons actually fill them. A basis of $k$ AOs gives $k$ MOs, and only the occupied ones contribute to the ground-state energy.

• The bonding/antibonding designation is not absolute. It depends on the sign of $\beta$: $\beta <0$ (typical for $\sigma$ overlap between $s$ or aligned $p$ orbitals) makes the in-phase combination bonding; $\beta >0$ (which can occur with certain $p$-orbital alignments) would reverse the labels. The variational principle still works; the names get swapped.

• "Bond order $=1$" does not always mean "covalent single bond as taught in gen-chem." In F2, the bond order is 1, but the bond is much weaker than in H2 (158 vs 436 kJ/mol) because F2's filled bonding orbitals are offset by lone-pair-like antibonding lobes that destabilise the bond.

• The $\sigma$-$\pi$ ordering switches between N2 and O2. In B2, C2, N2 the ${\pi }_u(2p)$ orbitals sit below the ${\sigma }_g(2p_z)$; in O2, F2, Ne2 the order reverses to ${\sigma }_g(2p_z)$ below ${\pi }_u(2p)$. The switch is caused by $2s$-$2p$ mixing, which is significant on the left of the second row (where $2s$ and $2p$ are close in energy) and small on the right (where $2s$ sits well below $2p$). This ordering shift is the reason different second-row diatomics need different MO diagrams drawn at the start.

• A "bonding" MO is not magically attractive. Its energy lowering compared to the AOs comes from the electrons spending more time in the internuclear region where they feel attraction from both nuclei at once. Antibonding orbitals have a node between the nuclei, so electrons there are excluded from the bonding region. The same orbital picture, with electron-electron repulsion turned on, becomes the multi-electron Hartree-Fock theory at Master tier.

Key theorem with proof [Intermediate+]

Theorem (Variational principle for the LCAO ansatz). Let $\hat{H}$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$, with discrete ground-state energy $E_0:=\inf \mathrm{spec}(\hat{H})$. Let $V\subset \mathcal{H}$ be a finite-dimensional subspace spanned by linearly independent vectors $\{{\varphi }_i{\}}_{i=1}^N$ with overlap matrix $S_{ij}=⟨{\varphi }_i|{\varphi }_j⟩$ and Hamiltonian matrix $H_{ij}=⟨{\varphi }_i|\hat{H}|{\varphi }_j⟩$. Then the smallest eigenvalue $E_{-}$ of the generalised eigenvalue problem

$$H\mathbf{c}=ES\mathbf{c},\mathbf{c}\in {\mathbb{C}}^N,$$

satisfies $E_{-}\ge E_0$, with equality if and only if the true ground state lies in $V$.

Proof. Any trial vector $\psi \in V$ may be written $\psi ={\sum }_i{c}_i{\varphi }_i$. The Rayleigh quotient is

$$\mathcal{R}[\psi ]=\frac{⟨\psi |\hat{H}|\psi ⟩}{⟨\psi |\psi ⟩}=\frac{{\mathbf{c}}^{†}H\mathbf{c}}{{\mathbf{c}}^{†}S\mathbf{c}}.$$

Stationary points of $\mathcal{R}$ are obtained by setting the gradient with respect to ${\mathbf{c}}^{†}$ to zero (the complex conjugate variation $c_i^{\ast }$ may be varied independently of $c_i$ since $\mathcal{R}$ is real-valued):

$$\frac{\partial }{\partial c_i^{\ast }}({\mathbf{c}}^{†}H\mathbf{c}-\mathcal{R}{\mathbf{c}}^{†}S\mathbf{c})=(H\mathbf{c}{)}_i-\mathcal{R}(S\mathbf{c}{)}_i=0,$$

which is the generalised eigenvalue equation $H\mathbf{c}=\mathcal{R}S\mathbf{c}$ with $\mathcal{R}=E$ at the stationary point. The stationary values are exactly the $N$ generalised eigenvalues $E_{-}=E_1\le E_2\le \cdots \le E_N$.

Since $S$ is positive-definite Hermitian, write $S=L{L}^{†}$ (Cholesky decomposition). Setting $\mathbf{d}:=L^{†}\mathbf{c}$ transforms the problem into the ordinary eigenvalue problem $L^{-1}H{L}^{-†}\mathbf{d}=E\mathbf{d}$ for the Hermitian matrix $L^{-1}H{L}^{-†}$, whose smallest eigenvalue $E_{-}$ is what we want.

For any normalised $\psi \in V$ ($⟨\psi |\psi ⟩=1$), the spectral theorem applied to $\hat{H}$ gives $⟨\psi |\hat{H}|\psi ⟩=\int \lambda d{\mu }_{\psi }(\lambda )\ge E_0$, where ${\mu }_{\psi }$ is the spectral measure of $\hat{H}$ on $\psi$. Therefore $\mathcal{R}[\psi ]\ge E_0$ for every $\psi \in V\setminus \{0\}$. Taking the infimum over $V$, the smallest stationary value $E_{-}$ — which equals ${\inf }_V\mathcal{R}$ by the Courant-Fischer characterisation of the smallest eigenvalue of a Hermitian matrix — also satisfies $E_{-}\ge E_0$.

Equality $E_{-}=E_0$ requires the infimum to be attained, which by the Courant-Fischer argument requires the ground-state eigenvector of $\hat{H}$ to lie in $V$. ∎

Corollary (Operational LCAO method). Diagonalising the generalised eigenvalue problem on the basis $\{{\varphi }_i\}$ yields $N$ approximate molecular orbitals with energies $E_1\le \cdots \le E_N$ and coefficient vectors ${\mathbf{c}}^{(k)}$. Increasing the basis size (more atomic orbitals per atom; polarisation and diffuse functions added) monotonically decreases the predicted ground-state energy and improves the approximation to the true MO.

The Courant-Fischer minimax statement extends the result above to higher MOs: $E_k$ approximates the $k$-th eigenvalue of $\hat{H}$ from above. This is the Hylleraas-Undheim theorem (1930), the higher-energy analogue of the variational principle.

Worked example: secular equation for H2 in the minimal $1s$ basis

With $\alpha =H_{AA}=H_{BB}$, $\beta =H_{AB}$, and $S=S_{AB}$, the secular equation is

$$\det \left(\begin{array}{cc}\alpha -E& \beta -ES\\ \beta -ES& \alpha -E\end{array}\right)=(\alpha -E{)}^2-(\beta -ES{)}^2=0.$$

Factoring: $(\alpha -E-\beta +ES)(\alpha -E+\beta -ES)=0$. The two roots are

$$E_{-}=\frac{\alpha +\beta }{1+S}({\sigma }_g,\text{ bonding}),E_{+}=\frac{\alpha -\beta }{1-S}({\sigma }_u^{\ast },\text{ antibonding}).$$

For typical numerical values at the H2 equilibrium internuclear distance $R_e\approx 74$ pm in a minimal Slater-type-orbital basis: $\alpha \approx -13.6$ eV (hydrogen 1s binding energy), $\beta \approx -19.2$ eV, $S\approx 0.59$. Substituting:

$$E_{-}=\frac{-13.6+(-19.2)}{1+0.59}=\frac{-32.8}{1.59}\approx -20.6\text{eV},$$ $$E_{+}=\frac{-13.6-(-19.2)}{1-0.59}=\frac{5.6}{0.41}\approx +13.7\text{eV}.$$

The bonding orbital sits below the atomic 1s ($\alpha =-13.6$ eV) by about 7 eV; the antibonding sits above by about 27 eV. The asymmetry — antibonding more destabilising than bonding is stabilising — is what kills He2: the four-electron filling pays more in the antibonding penalty than it gains from the bonding lowering.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not yet cover quantum chemistry. The closest layers are:

• Mathlib.LinearAlgebra.Matrix.Hermitian: Hermitian matrices and the spectral theorem in finite dimension — enough to state the secular equation $\det (H-ES)=0$ on a fixed basis.
• Mathlib.Analysis.InnerProductSpace.Basic: inner product spaces, used to state matrix elements $H_{ij}=⟨{\varphi }_i|\hat{H}|{\varphi }_j⟩$.
• Mathlib.LinearAlgebra.Matrix.PosDef: positive-definite Hermitian matrices, used for the overlap matrix $S$ and its Cholesky decomposition.

The LCAO ansatz, the variational principle on a finite trial subspace, the Slater determinant antisymmetriser, the Hartree-Fock equations, and the Roothaan-Hall SCF iteration are not in Mathlib. The formalisation pathway is laid out in lean_mathlib_gap in the frontmatter.

lean_status: none reflects this gap; no lean_module ships with this unit. The aggregated lean_status: none units in §14 become a Mathlib contribution roadmap as that section grows.

Hartree-Fock and the many-electron extension [Master]

The intermediate-tier LCAO method treats a single electron in the field of fixed nuclei plus an effective one-electron potential. The realistic problem is many-electron: $N$ interacting electrons in the field of the nuclei, governed by the molecular Hamiltonian

$$\hat{H}=-\sum _{i=1}^N\frac{{\hslash }^2}{2m_e}{\nabla }_i^2-\sum _{i,A}\frac{Z_A{e}^2}{4\pi {\epsilon }_0{r}_{iA}}+\sum _{i<j}\frac{e^2}{4\pi {\epsilon }_0{r}_{ij}}+\sum _{A<B}\frac{Z_A{Z}_B{e}^2}{4\pi {\epsilon }_0{R}_{AB}},$$

where $A,B$ index nuclei and $i,j$ electrons. The Born-Oppenheimer approximation treats the nuclear coordinates as fixed parameters and the last (nuclear-repulsion) term as a constant; the electronic problem is then to find the ground state of the first three terms acting on $\Psi ({\mathbf{r}}_1,{\sigma }_1,\dots ,{\mathbf{r}}_N,{\sigma }_N)$, antisymmetric under exchange of any electron pair (Pauli exclusion at the multi-electron level 12.01.02 pending).

Slater determinant. The single most important approximation is to write $\Psi$ as a single antisymmetric product of $N$ one-electron spin-orbitals ${\chi }_a={\psi }_a(\mathbf{r})\otimes |{\sigma }_a⟩$:

$$\Psi ({\mathbf{x}}_1,\dots ,{\mathbf{x}}_N)=\frac{1}{\sqrt{N!}}\det \left(\begin{array}{ccc}{\chi }_1({\mathbf{x}}_1)& \cdots & {\chi }_N({\mathbf{x}}_1)\\ ⋮& & ⋮\\ {\chi }_1({\mathbf{x}}_N)& \cdots & {\chi }_N({\mathbf{x}}_N)\end{array}\right),$$

where $\mathbf{x}:=(\mathbf{r},\sigma )$ collects spatial and spin coordinates. Antisymmetry is built in by the determinant. This Slater determinant is the simplest interacting fermionic wavefunction; the spin-statistics requirement of antisymmetry forces it on us, with the QFT-level justification given in 12.01.02 pending and its sequel.

Hartree-Fock equations. The variational principle, applied to the trial wavefunction class of single Slater determinants, yields equations for the orbitals ${\chi }_a$:

$$\hat{F}{\chi }_a={\epsilon }_a{\chi }_a,$$

where the Fock operator $\hat{F}$ is

$$\hat{F}=\hat{h}+\sum _b({\hat{J}}_b-{\hat{K}}_b),$$

with $\hat{h}$ the one-electron operator (kinetic + nuclear attraction), ${\hat{J}}_b$ the Coulomb operator capturing the classical electrostatic field of the other electrons, and ${\hat{K}}_b$ the exchange operator — a nonlocal operator with no classical analogue, arising from the antisymmetry of the Slater determinant. The orbital energies ${\epsilon }_a$ are the eigenvalues of $\hat{F}$.

Self-consistent field (SCF) iteration. The Fock operator depends on the orbitals through $\hat{J}$ and $\hat{K}$, making the equation nonlinear. The standard solution algorithm is the Roothaan-Hall procedure: choose an initial guess for the orbital coefficients on an atomic-orbital basis (the LCAO basis), build $\hat{F}$, solve the generalised eigenvalue problem $F\mathbf{c}=\epsilon S\mathbf{c}$ on the basis, get new orbitals, rebuild $\hat{F}$, iterate to self-consistency. The procedure converges (under suitable damping / level-shifting) to the variational minimum within the chosen basis.

Koopmans' theorem. Within the Hartree-Fock approximation and assuming the orbitals are frozen on ionisation (the frozen-orbital approximation), the canonical orbital energy ${\epsilon }_a$ equals the negative of the ionisation potential from that orbital (Koopmans 1934). For the highest occupied molecular orbital (HOMO), $-{\epsilon }_{\text{HOMO}}$ approximates the first ionisation energy; for the lowest unoccupied (LUMO), $-{\epsilon }_{\text{LUMO}}$ approximates the electron affinity (with sign caveats). The approximation neglects orbital relaxation and electron correlation; for the second-row diatomics it gives ionisation energies typically accurate to 1–2 eV.

Brillouin's theorem. The Hartree-Fock ground-state Slater determinant $|{\Psi }_0⟩$ has vanishing matrix element of the full Hamiltonian with any singly-excited determinant: $⟨{\Psi }_0|\hat{H}|{\Psi }_a^r⟩=0$ where ${\Psi }_a^r$ replaces occupied orbital $a$ with virtual orbital $r$. This is the reason single excitations alone cannot improve the HF energy; configuration interaction (CI) corrections must start from double excitations.

Basis sets. In practice the atomic-orbital basis $\{{\varphi }_i\}$ is either Slater-type orbitals (STOs), of the form $r^{n-1}{e}^{-\zeta r}{Y}_{\ell }^m(\theta ,\phi )$, which are physically correct for hydrogen-like atoms but produce difficult-to-evaluate two-electron integrals, or Gaussian-type orbitals (GTOs), $r^{n-1}{e}^{-\alpha r^2}{Y}_{\ell }^m$, which are not physically correct (no nuclear cusp, wrong asymptotic decay) but yield integrals expressible in closed form. Boys (1950) introduced GTOs into quantum chemistry precisely for this computational advantage. Modern basis sets (the Pople 6-31G family, the Dunning correlation-consistent cc-pVXZ family, the Karlsruhe def2 family) are linear combinations of GTOs designed to mimic STO behaviour while retaining computational tractability.

The HF approximation's limitation. Hartree-Fock is the exact solution of the molecular Schrödinger equation within the single-determinant approximation. The difference between the HF energy and the true energy (at fixed basis) is called the correlation energy:

$$E_{\text{corr}}=E_{\text{exact}}-E_{\text{HF}}.$$

It is always negative (the true energy is lower than HF). Correlation accounts for the instantaneous Coulomb avoidance between electrons, which a single Slater determinant cannot capture. For chemically meaningful accuracy on bond energies, post-HF methods are mandatory.

Post-Hartree-Fock methods. Four broad families repair the missing correlation:

• Configuration interaction (CI) expands $\Psi$ as a sum of Slater determinants with the HF determinant as the leading term and excited determinants as corrections. Full CI in a given basis is variational and exact within that basis; truncated CI (CISD = singles + doubles) is variational but size-inconsistent.
• Many-body perturbation theory (Møller-Plesset, MP$n$) treats electron correlation perturbatively. MP2 is the most common — second-order correction; size-consistent but not variational.
• Coupled cluster (CC) writes $\Psi =e^{\hat{T}}|{\Psi }_0⟩$ with $\hat{T}$ a sum of excitation operators (singles ${\hat{T}}_1$, doubles ${\hat{T}}_2$, ...); the CCSD(T) variant (singles + doubles + perturbative triples) is the modern gold standard for high-accuracy molecular calculations.
• Density functional theory (DFT) sidesteps the wavefunction altogether and writes the ground-state energy as a functional of the electron density $\rho (\mathbf{r})$. The Hohenberg-Kohn theorem (1964) guarantees such a functional exists and is universal; the Kohn-Sham equations (1965) recast the problem as a single-determinant calculation with an exchange-correlation potential $v_{\text{xc}}[\rho ]$ replacing the exact correlation. Modern functionals (B3LYP, PBE, M06, ωB97X-D) are constructed by mixing exact-exchange fractions, empirical parameters, and density-gradient terms; their accuracy on bond energies, reaction barriers, and excited states varies by chemistry.

The MO picture survives, with reinterpretation: HF orbitals (the canonical Hartree-Fock single-determinant building blocks) and Kohn-Sham orbitals (the analogous DFT objects) have similar shapes for typical molecules but conceptually different status — the HF orbital is a one-particle wavefunction in a mean field; the KS orbital is a fictitious orbital of an auxiliary non-interacting system that reproduces the true density. For chemistry intuition both serve, and the differences matter for spectroscopy and excited states.

Symmetry-adapted linear combinations and the role of group theory [Master]

For polyatomic molecules the LCAO basis becomes large and the secular determinant unwieldy. Symmetry-adapted linear combinations (SALCs) exploit the molecular point group to block-diagonalize the Fock matrix: the matrix $F$ is invariant under the group's action on the basis, so $F$ commutes with each representation operator $D(g)$, hence Schur's lemma 07.01.02 forces $F$ to be a scalar on each irreducible isotypic subspace.

For a homonuclear diatomic in $D_{\infty h}$, the irreducible representations relevant to second-row chemistry are ${\sigma }_g^{+},{\sigma }_u^{+},{\pi }_g,{\pi }_u$ (and higher angular-momentum labels $\delta ,\varphi ,\dots$ for $d$, $f$, $\dots$ orbital combinations, irrelevant to second-row). The $2s$ and $2p_z$ atomic orbitals span ${\sigma }_g+{\sigma }_u$ on each side, so after symmetry adaptation the four $\sigma$-type MOs are ${\sigma }_g(2s),{\sigma }_u^{\ast }(2s),{\sigma }_g(2p_z),{\sigma }_u^{\ast }(2p_z)$ — all mutually orthogonal by symmetry, with no zero off-diagonal block. The $2p_x,2p_y$ atomic orbitals span ${\pi }_u+{\pi }_g$, giving the two-fold degenerate ${\pi }_u$ bonding and ${\pi }_g^{\ast }$ antibonding sets.

The character-theoretic tool that makes this systematic is the projection-operator formula from 07.01.04:

$${\hat{P}}^{(\Gamma )}=\frac{d_{\Gamma }}{|G|}\sum _{g\in G}{\chi }^{(\Gamma )\ast }(g)D(g),$$

where $\Gamma$ ranges over irreducible representations, ${\chi }^{(\Gamma )}$ is the character, and $D(g)$ acts on the AO basis. Applying ${\hat{P}}^{(\Gamma )}$ to each AO and discarding linear dependencies yields the SALCs that transform as $\Gamma$.

For molecules with finite point groups (water, ammonia, benzene, methane, ferrocene, ...) the same machinery applies, and the symmetry-block structure of $F$ replaces a single huge eigenvalue problem with many small ones. The full machinery — projection operators, descent in symmetry, character tables, correlation tables — is the core of the inorganic-chemistry application of group theory (cross-cite 16.02.01, pending).

Beyond the homonuclear case and the MO/VB reconciliation [Master]

The MO picture and the valence-bond (VB) picture of Heitler-London-Pauling are different starting points for the same problem and converge in the limit of large basis + many correlations. For H2 at equilibrium $R_e$, HF-MO is qualitatively right; far past dissociation, $R\to \infty$, single-determinant HF gives the wrong dissociation limit because it forces the two electrons into the same spatial orbital ${\sigma }_g(1s)$ rather than allowing the physically correct ${\varphi }_A(1){\varphi }_B(2)$ + ${\varphi }_A(2){\varphi }_B(1)$ asymptote with electrons on different atoms. Generalised valence bond (GVB) repairs this by allowing the bonding-orbital pair to spin-couple as a singlet of two localised orbitals; full CI gives the same fix automatically. The lesson for the qualitative MO/VB tension: MO is correct at equilibrium and over compressions; VB is correct at large separation; both miss "static correlation" features that demand multireference methods.

For asymmetric (heteronuclear) diatomics, the homonuclear $g/u$ labels are no longer available — inversion symmetry is broken. The MO ordering becomes determined by the on-site energies ${\alpha }_A\ne {\alpha }_B$ and the coupling $\beta$, as treated in Exercise 10. In the strongly polar limit $|{\alpha }_A-{\alpha }_B|\gg |\beta |$, the bonding MO is dominantly atomic-orbital $B$ (the more electronegative atom), the antibonding MO is dominantly $A$, and the bond is polar covalent in the textbook sense — a continuous interpolation between covalent (symmetric LCAO mixing) and ionic ($A^{+}{B}^{-}$) bonding. This single picture unifies what gen-chem teaches as separate covalent/polar/ionic categories.

Connections [Master]

• Variational principle and the secular equation rest on the spectral theorem for Hermitian operators on finite-dimensional inner-product spaces — see vector spaces 01.01.03 and eigenvalues 01.01.08. The infinite-dimensional generalisation (HF as a Fréchet-derivative variational problem on the Stiefel manifold of orbital coefficient matrices) is in functional analysis 02.11.08.

• Pauli exclusion and the Slater determinant are direct consequences of the spin-statistics theorem — the QM-side foundation is in 12.01.02 pending, whose Master tier mentions the QFT-level argument that fermions of half-integer spin must have antisymmetric many-body wavefunctions. The Slater-determinant antisymmetriser is the simplest finite-dimensional model of fermionic Fock space.

• Group representations and character tables organise the symmetry adaptation: each irreducible representation of the molecular point group 07.01.01 labels a subset of MOs, and the character orthogonality relation 07.01.04 underlies the projection-operator formula for SALCs.

• The atomic-orbital basis itself — the ${\varphi }_i$ that go into the LCAO sum — comes from quantum chemistry of atoms (§14.04, pending). Atomic Hartree-Fock orbitals, hydrogenic orbitals, and STO/GTO parameterisations are inputs to the diatomic MO problem and outputs of the multi-electron atom theory.

• Spectroscopy of diatomics — the electronic spectra of N2, O2, CO, etc. — interpret transitions between the MOs of this unit. The selection rules and intensities flow from time-dependent perturbation theory and the dipole transition operator (§14.12, pending), feeding back into the electromagnetic-field treatment of light-matter interaction (§10 EM).

• Reciprocal hook to Stern-Gerlach 12.01.02 pending: the spin-1/2 nature of the electron and the Pauli-exclusion-driven pairing in MOs both descend from that physics-side foundation. The hook is proposed until a physics-side reviewer audits the cross-domain semantics.

• Forward to polyatomic MOs: Hückel theory for aromatic systems, the Hoffmann extended-Hückel method, and Woodward-Hoffmann rules for pericyclic reactions [15.08.NN, pending] all build on the diatomic MO foundation set here.

• Forward to inorganic chemistry: crystal field theory and ligand field theory [16.03.02, pending] treat $d$-orbital splittings in transition-metal complexes as a special case of MO theory in a low-symmetry environment. The MO/LFT picture is the modern unification of what older texts split into crystal-field and ligand-field accounts.

• DFT as an alternative paradigm: the Kohn-Sham orbitals of density functional theory occupy the conceptual slot of the HF orbitals here, but with the exchange operator replaced by an exchange-correlation functional. Most modern computational chemistry runs on DFT rather than HF; the qualitative MO picture survives the transition.

• Bio-side hook on heme chemistry [17.04.01, proposed]: the binding of O2 to a Fe(II) heme is treated chemically as a coordination event between the O2 ${\pi }_g^{\ast }$ orbital and Fe d-orbitals, with the spin state of the bound O2-Fe complex (singlet vs triplet) determined by the relative energies set in this unit's MO diagram.

Historical & philosophical context [Master]

Molecular orbital theory has two founding moments. Hund (1926, 1928) ^{[Hund 1926]} and Mulliken (1928, 1932) ^{[Mulliken 1928]} independently developed the picture in which molecular electrons occupy delocalised orbitals labelled by molecular quantum numbers — Mulliken coined the term molecular orbital and constructed the diatomic correlation diagrams that survive almost unchanged in modern textbooks. Lennard-Jones (1929) ^{[Lennard-Jones 1929]} gave the LCAO ansatz its working form for H2${}^{+}$. Roothaan (1951) ^{[Roothaan 1951]} cast multi-electron Hartree-Fock onto an algebraic finite basis — the Roothaan-Hall equations — and made systematic computational quantum chemistry possible.

The competing valence-bond (VB) picture was developed by Heitler and London (1927), Slater (1931), and Pauling (1939) and held the dominant pedagogical position until the 1950s; the rise of MO theory was driven both by computational advantages (the SCF method's algebraic structure is more amenable to general-purpose code than VB's resonance bookkeeping) and by qualitative successes — the prediction of O2 paramagnetism is the most celebrated, but the systematic correlation diagrams across the periodic table that MO theory makes natural are equally important.

Density functional theory entered chemistry through Hohenberg-Kohn (1964) ^{[Hohenberg-Kohn 1964]} and Kohn-Sham (1965). Kohn shared the 1998 Nobel Prize in Chemistry with John Pople for the development of computational quantum chemistry; today DFT-based methods are the dominant production technique for chemical-scale electronic structure calculations.

The orbital ontology question — is a molecular orbital a real entity or a calculational device? — remains live in philosophy of chemistry. Hund himself emphasised that an MO is a one-electron approximation, with no precise correspondence to any feature of the true many-electron wavefunction. The Slater determinant builds many-electron states from one-electron orbitals, but it is invariant under unitary mixing of the occupied orbitals among themselves: the canonical HF orbitals one gets out of the SCF procedure are one of infinitely many orthogonal equivalent sets that give the same Slater determinant. So "the ${\sigma }_g(2s)$ orbital of N2" is more properly "any orbital in the occupied subspace that diagonalises the Fock operator with the canonical eigenvalue." Localised orbitals (Boys, Pipek-Mezey, Edmiston-Ruedenberg constructions) and natural orbitals (eigenvectors of the one-particle density matrix) offer alternative choices that emphasise different physical features. The reality status of "the orbital" is therefore subtler than gen-chem pedagogy suggests; the relevant phil-of-chemistry literature is Hoffmann-Laszlo-Schummer (2007) and Scerri's The Periodic Table (2007).

Bibliography [Master]

Primary literature (cite when used; not all currently in reference/):

• Hund, F., "Zur Deutung der Molekelspektren I, II, III," Z. Phys. 36 (1926), 657; 40 (1927), 742; 51 (1928), 759.

• Mulliken, R. S., "The assignment of quantum numbers for electrons in molecules I, II, III," Phys. Rev. 32 (1928), 186; 41 (1932), 49; 43 (1933), 279.

• Lennard-Jones, J. E., "The electronic structure of some diatomic molecules," Trans. Faraday Soc. 25 (1929), 668.

• Heitler, W. & London, F., "Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik," Z. Phys. 44 (1927), 455.

• Slater, J. C., "The theory of complex spectra," Phys. Rev. 34 (1929), 1293; "Molecular energy levels and valence bonds," Phys. Rev. 38 (1931), 1109.

• Pauling, L., The Nature of the Chemical Bond (Cornell University Press, 1939; 3rd ed. 1960).

• Boys, S. F., "Electronic wavefunctions I. A general method of calculation for the stationary states of any molecular system," Proc. R. Soc. A 200 (1950), 542.

• Roothaan, C. C. J., "New developments in molecular orbital theory," Rev. Mod. Phys. 23 (1951), 69.

• Koopmans, T., "Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den einzelnen Elektronen eines Atoms," Physica 1 (1934), 104.

• Brillouin, L., Actualités Scientifiques et Industrielles No. 159 (1934).

• Hohenberg, P. & Kohn, W., "Inhomogeneous electron gas," Phys. Rev. 136 (1964), B864.

• Kohn, W. & Sham, L. J., "Self-consistent equations including exchange and correlation effects," Phys. Rev. 140 (1965), A1133.

• Atkins, P. & de Paula, J., Physical Chemistry, 11th ed. (Oxford University Press, 2018).

• McQuarrie, D. A., Quantum Chemistry, 2nd ed. (University Science Books, 2008).

• Levine, I. N., Quantum Chemistry, 7th ed. (Pearson, 2013).

• Szabo, A. & Ostlund, N. S., Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Dover, 1996).

• Housecroft, C. E. & Sharpe, A. G., Inorganic Chemistry, 5th ed. (Pearson, 2018).

• Scerri, E. R., The Periodic Table: Its Story and Its Significance (Oxford University Press, 2007).

• Hoffmann, R., Laszlo, P. & Schummer, J., eds., Reading the Atom: Philosophical Reflections on Chemistry (HYLE special issue, 2007).

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Wave 1 chemistry seed unit, produced manually 2026-05-18 (per docs/plans/CHEMISTRY_PLAN.md §6 — manual-first pattern-setter for §14–16). All four cross-domain hooks_out targets are proposed; promotion to confirmed awaits cross-domain reviewer audit. Hard prereqs cite only shipped math units (linear algebra, eigenvalue theory, group representations, characters); the §12 QM Stern-Gerlach unit is cited as a reciprocal hooks_out proposed link rather than a hard prereq, because 12.01.02 is currently status: draft and is not registered in manifests/deps.json pending list — see report for the proposed deps.json addition that would let it become a confirmed cross-domain prereq once 12.01.02 ships. Status remains draft pending Tyler's review and external quantum-chemistry-reviewer attestation per CHEMISTRY_PLAN §7.