Hartree-Fock method (concept); DFT (concept) — Chemistry notes

1. WHY we need approximations

WHY it's hard: the term $\dfrac{1}{r_{ij}}$ couples electron $i$ and $j$ . You cannot write $\Psi$ as a simple product because moving electron 1 changes the force on electron 2. The wavefunction $\Psi(\mathbf x_1,\dots,\mathbf x_N)$ depends on $3N$ spatial + $N$ spin coordinates — exponential blow-up.

2. Hartree–Fock: the average-field trick

2.1 The Slater determinant (WHAT the wavefunction is)

WHY a determinant? Electrons are fermions: $\Psi$ must change sign when two are swapped (Pauli antisymmetry). Swapping two electrons = swapping two rows of a determinant = flips the sign. Bonus: if two electrons share the same spin-orbital, two columns are equal → determinant $=0$ . That is the Pauli exclusion principle, baked in automatically.

2.2 The HF equations (HOW we get them)

We minimize the energy $E=\langle\Psi_{HF}|\hat H|\Psi_{HF}\rangle$ subject to orthonormal orbitals (Lagrange multipliers $\varepsilon_i$ ). The variational result is a set of one-electron eigenvalue equations:

Where does $\hat J - \hat K$ come from? When you expand $\langle\Psi_{HF}|\sum 1/r_{ij}|\Psi_{HF}\rangle$ , every pair gives two contributions: $\underbrace{\iint \frac{|\chi_i(1)|^2|\chi_j(2)|^2}{r_{12}}}_{\text{Coulomb }J_{ij}}-\underbrace{\iint \frac{\chi_i^*(1)\chi_j^*(2)\chi_j(1)\chi_i(2)}{r_{12}}}_{\text{Exchange }K_{ij}}$ The exchange integral only survives between same-spin electrons (spins integrate to 0 otherwise) — that's why same-spin electrons keep apart ("Fermi hole").

2.3 What HF misses: correlation energy

3. DFT: density is everything

WHY HK1 is believable (Feynman-style proof sketch): Suppose two different potentials $v_1\ne v_2$ gave the same $\rho$ . Plug each ground state into the other's Hamiltonian; the variational principle gives two strict inequalities that add to $E_1+E_2 < E_1+E_2$ — a contradiction. So the map $\rho\to v$ is unique.

3.1 The Kohn–Sham trick (HOW DFT becomes solvable)

The kinetic-energy functional of $\rho$ is unknown. Kohn & Sham's fix: introduce a fictitious system of non-interacting electrons with the same density $\rho$ as the real one.

The energy splits as: $E[\rho]=T_s[\rho]+\int v_{ext}\rho + J[\rho] + \underbrace{E_{xc}[\rho]}_{\text{exchange–correlation}}$

$T_s$ : kinetic energy of the non-interacting electrons (computed exactly from $\phi_i$ ),
$J[\rho]$ : classical Coulomb,
$E_{xc}$ : everything unknown swept into one term — exchange + correlation + kinetic correction.

Common mistake HF vs DFT — the key contrast

	Hartree–Fock	Kohn–Sham DFT
Central object	wavefunction (Slater det.)	density $\rho(\mathbf r)$
Exchange	exact	approximate (in $E_{xc}$ )
Correlation	missing	approximately included
Cost (formal)	$\sim N^4$	$\sim N^3$

4. Worked conceptual examples

Recall Feynman: explain to a 12-year-old

Imagine a crowded room where everyone repels everyone (like same magnets). Figuring out exactly how everyone dodges everyone is hopeless. Hartree–Fock says: "Pretend each person just feels the average crowd, not each individual." Easier! But it misses people swerving around specific others — that missing swerving is "correlation." DFT says something wilder: "I don't need to track every person; I just need the map of how crowded each spot is (the density). That map secretly tells me everything." Both turn an impossible group problem into a one-person-at-a-time problem you can actually solve by repeating until nothing changes.

Flashcards

Why is the many-electron Schrödinger equation unsolvable exactly?

The

1/r_{ij}

electron–electron repulsion couples all electrons, so

\Psi

cannot be separated into independent one-electron parts.

What is the central approximation of Hartree–Fock?

Each electron moves in the average (mean) field of all the others, and

\Psi

is a single Slater determinant.

Why use a Slater determinant?

It is automatically antisymmetric (sign flips on swapping two electrons = swapping rows), enforcing Pauli exclusion (two identical orbitals → determinant = 0).

What is the exchange term

K

physically?

A purely quantum-mechanical interaction with no classical analogue, arising from antisymmetry; it keeps same-spin electrons apart (Fermi hole) and lowers their energy.

What does SCF mean and why is it iterative?

Self-Consistent Field: the Fock operator depends on the orbitals it produces, so you iterate guess→build→solve until orbitals stop changing.

Why isn't

E_{HF}

just the sum of orbital energies?

Summing all

\varepsilon_i

double-counts every electron–electron pair repulsion, so

E=\sum\varepsilon_i-\tfrac12\sum(J_{ij}-K_{ij})

Define correlation energy.

E_{corr}=E_{exact}-E_{HF}<0

; the energy HF misses because electrons can't instantaneously dodge each other in a mean field.

State Hohenberg–Kohn Theorem 1.

The ground-state electron density

\rho(\mathbf r)

uniquely determines the external potential and hence all properties;

E

is a functional

E[\rho]

State Hohenberg–Kohn Theorem 2.

A variational principle: the true ground-state density minimizes

E[\rho]

, and

E[\rho]\ge E_0

for any trial density.

What is the Kohn–Sham trick?

Replace the real interacting electrons with a fictitious non-interacting system having the same density, making the kinetic energy computable exactly via orbitals

\phi_i

What is

E_{xc}

and why is it the problem child of DFT?

The exchange–correlation functional collects all unknown contributions; DFT is exact only if

E_{xc}

were known, but it must be approximated (LDA, GGA, hybrids).

Key difference: HF vs DFT on exchange/correlation?

HF has exact exchange but no correlation; KS-DFT has approximate exchange but approximately includes correlation.

Connections

Schrödinger Equation — the exact equation we approximate.
Born-Oppenheimer Approximation — fixes nuclei first.
Pauli Exclusion Principle — enforced by the Slater determinant.
Variational Principle — basis of both HF minimization and HK2.
Basis Sets (STO, GTO) — how orbitals are represented numerically.
Electron Correlation Methods (MP2, CI, CCSD) — post-HF fixes for $E_{corr}$ .
Molecular Orbital Theory — HF orbitals are the canonical MOs.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek atom ya molecule me jab kai electrons hote hain, to har electron baaki sab ko repel karta hai (same charge). Is wajah se exact Schrödinger equation solve karna impossible hai — kyunki ek electron ka motion doosre pe depend karta hai, sab ek doosre me uljhe hue hain. Isko bolte hain electron correlation, aur yahi sabse bada sirdard hai.

Hartree–Fock ka jugaad simple hai: maan lo har electron ko baaki electrons ka sirf average cloud dikhta hai, individual electron nahi. Isse problem ek-ek electron ka ban jaata hai jo solve ho sakta hai. Wavefunction ko ek Slater determinant ke roop me likhte hain — yeh automatically Pauli exclusion principle follow karta hai (do electron same orbital me ho to determinant zero). Lekin Fock operator khud orbitals pe depend karta hai, to SCF loop chalate hain: guess karo, banao, solve karo, repeat — jab tak orbitals change hona band na ho jaayein. HF ka problem: average field ki wajah se electrons ek doosre ko instantly dodge nahi kar paate, to thodi energy (correlation energy) miss ho jaati hai — chhoti hai par chemistry me bahut important.

DFT ne ek bilkul alag, genius idea diya: wavefunction ko bhool jao! Hohenberg–Kohn theorem kehta hai ki ground-state ki saari information sirf electron density $\rho(\mathbf r)$ me chhupi hai — jo sirf 3 variables ka function hai (wavefunction ke $3N$ ki jagah). Kohn–Sham ne trick lagayi: ek fake non-interacting system banao jiska density same ho, aur saari unknown cheezein (exchange + correlation) ek term $E_{xc}$ me daal do. DFT theoretically exact hai agar $E_{xc}$ pata ho — par woh pata nahi, isliye approximations (LDA, GGA, B3LYP) use karte hain. Bottom line: HF me exchange exact par correlation gayab; DFT me correlation approximately included aur computation sasta — isliye aaj DFT chemistry me sabse zyada use hota hai.

Test yourself — Physical Chemistry (Advanced)

Connections