Hartree-Fock method (concept); DFT (concept)

5.1.3 · Chemistry › Physical Chemistry (Advanced)

1. WHY hume approximations ki zaroorat hai

WHY yeh mushkil hai: $\frac{1}{r _{ij}}$ wala term electron $i$ aur $j$ ko couple karta hai. Aap $Ψ$ ko ek simple product ki tarah nahi likh sakte kyunki electron 1 ko move karne se electron 2 par force badal jaati hai. Wavefunction $Ψ (x_{1}, \dots, x_{N})$ $3 N$ spatial aur $N$ spin coordinates par depend karti hai — exponential blow-up.

2. Hartree–Fock: average-field trick

2.1 Slater determinant (wavefunction KIYA hai)

WHY determinant? Electrons fermions hote hain: jab do ko swap karo toh $Ψ$ ka sign badalna chahiye (Pauli antisymmetry). Do electrons ko swap karna = determinant ki do rows ko swap karna = sign flip. Bonus: agar do electrons ek hi spin-orbital share karein, to do columns equal ho jaate hain → determinant $= 0$ . Yahi hai Pauli exclusion principle, automatically baked in.

2.2 HF equations (HOW hum inhe lete hain)

Hum energy $E = ⟨ Ψ_{H F} ∣ \hat{H} ∣ Ψ_{H F} ⟩$ ko minimize karte hain orthonormal orbitals ke subject mein (Lagrange multipliers $ε_{i}$ ). Variational result ek set of one-electron eigenvalue equations hai:

$\hat{J} - \hat{K}$ kahan se aata hai? Jab aap $⟨ Ψ_{H F} ∣ \sum 1/ r_{ij} ∣ Ψ_{H F} ⟩$ expand karte hain, har pair do contributions deta hai: $Coulomb J_{ij} \iint \frac{∣ χ _{i} ( 1 ) ∣ ^{2} ∣ χ _{j} ( 2 ) ∣ ^{2}}{r _{12}} - Exchange K_{ij} \iint \frac{χ _{i}^{*} ( 1 ) χ _{j}^{*} ( 2 ) χ _{j} ( 1 ) χ _{i} ( 2 )}{r _{12}}$ Exchange integral sirf same-spin electrons ke beech survive karta hai (warna spins integrate hokar 0 dete hain) — isliye same-spin electrons alag rehte hain ("Fermi hole").

2.3 HF kya miss karta hai: correlation energy

3. DFT: density hi sab kuch hai

WHY HK1 believable hai (Feynman-style proof sketch): Maano do alag potentials $v_{1} \neq = v_{2}$ ne ek hi $ρ$ diya. Har ground state ko doosre ke Hamiltonian mein daalo; variational principle do strict inequalities deta hai jo add hone par $E_{1} + E_{2} < E_{1} + E_{2}$ banate hain — ek contradiction. Toh map $ρ \to v$ unique hai.

3.1 Kohn–Sham trick (HOW DFT solvable banta hai)

$ρ$ ka kinetic-energy functional unknown hai. Kohn aur Sham ka fix: ek fictitious system of non-interacting electrons introduce karo jisme real system jaisi hi density $ρ$ ho.

Energy is split karti hai: $E [ρ] = T_{s} [ρ] + \int v_{e x t} ρ + J [ρ] + exchange-correlation E_{x c} [ρ]$

$T_{s}$ : non-interacting electrons ki kinetic energy (bilkul exactly $ϕ_{i}$ se compute hoti hai),
$J [ρ]$ : classical Coulomb,
$E_{x c}$ : sab kuch jo unknown hai ek term mein daal diya — exchange + correlation + kinetic correction.

Common mistake HF vs DFT — key contrast

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

4. Worked conceptual examples

Recall Feynman: ek 12-saal ke bacche ko samjhao

Ek bheed-bhaad wale kamre ki kalpana karo jahan sab ek doosre ko repel karte hain (jaise same magnets). Exactly pata karna ki har koi kaise bach-bachkar chalta hai, yeh impossible hai. Hartree–Fock kehta hai: "Pretend karo ki har aadmi sirf average bheed feel karta hai, na ki kisi specific aadmi ko." Aasaan! Lekin yeh miss karta hai ki log kisi khaas ke around kaise swerve karte hain — yeh missing swerving hi "correlation" hai. DFT kuch aur hi kamaal ki baat kehta hai: "Mujhe har aadmi ko track nahi karna; mujhe bas kitni bheed har jagah hai uska map chahiye (the density). Woh map chhup ke sab kuch bata deta hai." Dono ek impossible group problem ko ek-ek-karke solve hone wale one-person problem mein badal dete hain, bas tab tak repeat karo jab tak kuch badalta nahi.

Flashcards

Many-electron Schrödinger equation exactly unsolvable kyun hai?

1/ r_{ij}

electron–electron repulsion saare electrons ko couple karti hai, isliye

Ψ

ko independent one-electron parts mein separate nahi kiya ja sakta.

Hartree–Fock ki central approximation kya hai?

Har electron baaki saare electrons ke average (mean) field mein move karta hai, aur

Ψ

ek single Slater determinant hai.

Slater determinant kyun use karte hain?

Yeh automatically antisymmetric hota hai (do electrons swap karne par sign flip hota hai = rows swap hoti hain), Pauli exclusion enforce karta hai (do identical orbitals → determinant = 0).

Exchange term

K

physically kya hai?

Ek purely quantum-mechanical interaction jiska koi classical analogue nahi, antisymmetry se aata hai; same-spin electrons ko alag rakhta hai (Fermi hole) aur unki energy lower karta hai.

SCF ka matlab kya hai aur yeh iterative kyun hai?

Self-Consistent Field: Fock operator un orbitals par depend karta hai jo woh produce karta hai, isliye aap iterate karte ho guess→build→solve jab tak orbitals change hona band na ho jaayein.

E_{H F}

sirf orbital energies ka sum kyun nahi hai?

Saare

ε_{i}

add karne par har electron–electron pair repulsion double-count hoti hai, isliye

E = \sum ε_{i} - \frac{1}{2} \sum (J_{ij} - K_{ij})

Correlation energy define karo.

E_{cor r} = E_{e x a c t} - E_{H F} < 0

; woh energy jo HF miss karta hai kyunki mean field mein electrons ek doosre ko instantaneously dodge nahi kar sakte.

Hohenberg–Kohn Theorem 1 batao.

Ground-state electron density

ρ (r)

external potential aur isliye saari properties uniquely determine karti hai;

E

ek functional

E [ρ]

hai.

Hohenberg–Kohn Theorem 2 batao.

Ek variational principle: sacchi ground-state density

E [ρ]

minimize karti hai, aur kisi bhi trial density ke liye

E [ρ] \geq E_{0}

Kohn–Sham trick kya hai?

Real interacting electrons ko ek fictitious non-interacting system se replace karo jisme real wali jaisi hi density ho, jisse kinetic energy exactly orbitals

ϕ_{i}

se compute ho sake.

E_{x c}

kya hai aur yeh DFT ka problem child kyun hai?

Exchange–correlation functional saare unknown contributions ikattha karta hai; DFT exact tabhi hoti jab

E_{x c}

known hoti, lekin ise approximate karna padta hai (LDA, GGA, hybrids).

HF vs DFT mein exchange/correlation par key difference kya hai?

HF mein exact exchange hai lekin correlation nahi; KS-DFT mein approximate exchange hai lekin correlation approximately included hai.

Connections

Schrödinger Equation — woh exact equation jise hum approximate karte hain.
Born-Oppenheimer Approximation — pehle nuclei fix karo.
Pauli Exclusion Principle — Slater determinant se enforce hota hai.
Variational Principle — HF minimization aur HK2 dono ka aadhar.
Basis Sets (STO, GTO) — orbitals numerically kaise represent hote hain.
Electron Correlation Methods (MP2, CI, CCSD) — $E_{cor r}$ ke liye post-HF fixes.
Molecular Orbital Theory — HF orbitals hi canonical MOs hote hain.