Correlations and symmetry in mean-field wavefunctions
Molecular orbital theory
Valence bond theory
Variational principle
E = \dfrac{\langle \psi|H|\psi\rangle }{\langle \psi|\psi\rangle}
Strategy:
Ground state minimizes
- Come up with a clever wavefunction parametrization
- Minimize its energy by varying parameters (we use Monte Carlo methods to do this)
Variational Monte Carlo
\ \dfrac{\langle \psi(\mathbf{p})|H|\psi(\mathbf{p})\rangle }{\langle \psi(\mathbf{p})|\psi(\mathbf{p})\rangle} = \sum_{\mathbf{n}} \rho_{\mathbf{n}} E_L[\mathbf{n}]
- Calculate energy: continuous time sampling
E_L[\mathbf{n}] = \sum_{\mathbf{m}} H_{\mathbf{n},\mathbf{m}} \frac{ \langle \mathbf{m}|\Psi(\mathbf{p})\rangle}{\langle \mathbf{n}|\Psi(\mathbf{p})\rangle}
overlap ratios
- Calculate energy gradient: by sampling
|\mathbf{n}\rangle =
local orbitals allow screening of distant excitations
\frac{\partial E(\mathbf{p})}{\partial\mathbf{p}}
- Change parameters: smart gradient descent
walker
\epsilon
- \( O(N^3) \) scaling, parallelizable
Restricted Hartree Fock (RHF)
Doubly occupied orbitals:
|\psi\rangle = c_{1\uparrow}^{\dagger}c_{1\downarrow}^{\dagger}c_{2\uparrow}^{\dagger}c_{2\downarrow}^{\dagger}\dots c_{\frac{N}{2}\uparrow}^{\dagger}c_{\frac{N}{2}\downarrow}^{\dagger}|0\rangle
c_{i\sigma}^{\dagger}=\sum_{l}\theta^{l}_{i}a^{\dagger}_{l\sigma}
atomic
molecular
\langle\mathbf{n}|\psi\rangle = \text{det}(\qquad)\text{det}(\qquad)
\(\theta\) slices
\psi=\mathcal{A}[\phi_1(r_{1\uparrow})\phi_1(r_{1\downarrow})\dots \phi_{N/2}(r_{N/2\uparrow})\phi_{N/2}(r_{N/2\downarrow})]
Shortcomings of restricted orbitals
- Doesn't correlate \(\uparrow\) and \(\downarrow\) electrons
\langle\rho_{i\uparrow}\rho_{j\downarrow}\rangle = \langle\rho_{i\uparrow}\rangle\langle\rho_{j\downarrow}\rangle,
\langle\rho_{i\uparrow}\rangle=\langle\rho_{i\downarrow}\rangle
- Doesn't break bonds correctly: \(\text{H}_2\) dissociation
Restricted
Exact
c_{1\uparrow}^{\dagger}c_{1\downarrow}^{\dagger}|0\rangle = (a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{1\downarrow}^{\dagger}a_{1\uparrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle
c_{1}^{\dagger}\qquad =\quad a_{1}^{\dagger}+a_{2}^{\dagger}
as \(d\rightarrow\infty\)
Unrestricted Hartree Fock (UHF)
Singly occupied orbitals:
|\psi\rangle = c_{1\uparrow}^{\dagger}c_{1\downarrow}^{\dagger}c_{2\uparrow}^{\dagger}c_{2\downarrow}^{\dagger}\dots c_{\frac{N}{2}\uparrow}^{\dagger}c_{\frac{N}{2}\downarrow}^{\dagger}|0\rangle
c_{i\sigma}^{\dagger}=\sum_{l}\theta^{l}_{i\sigma}a^{\dagger}_{l\sigma}
\langle\mathbf{n}|\psi\rangle = \text{det}(\qquad)\text{det}(\qquad)
- Correlates \(\uparrow\) and \(\downarrow\) electrons
- Breaks bonds correctly (in many cases)
Unrestricted
Exact
c_{1\uparrow}^{\dagger}=a_{1\uparrow}^{\dagger},\ c_{1\downarrow}^{\dagger}=a_{2\downarrow}^{\dagger}
as \(d\rightarrow\infty\)
Broken symmetry
- Hamiltonian has spin symmetry: invariant under spin rotations
[H, \mathbf{S}] = 0
The exact ground state is a spin eigenstate.
- The restricted wavefunction is a spin eigenstate, the unrestricted wavefunction is not. It breaks \( S^2 \) symmetry.
- The unrestricted wavefunction is an \( S_z \) eigenstate. We can break this symmetry as well.
- For an approximate variational wavefunction, symmetry is a constraint that can only raise its energy. In general, we can break all symmetries to remove constraints.
Generalized Hartree Fock (GHF)
|\psi\rangle = c_{1}^{\dagger}c_{2}^{\dagger}\dots c_{N}^{\dagger}|0\rangle
c_{i}^{\dagger}=\sum_{l,\sigma}\theta^{l\sigma}_{i}a^{\dagger}_{l\sigma}=\sum_{l}\left(\theta^{l\uparrow}_{i}a^{\dagger}_{l\uparrow}+\theta^{l\downarrow}_{i}a^{\dagger}_{l\downarrow}\right)
\langle\mathbf{n}|\psi\rangle = \text{det}(\qquad)
Orbitals are not \( S_z \) eigenstates, spin and space entangled
Frustrated systems:
Restore broken symmetries
- We can restore the broken symmetry by projecting onto the desired symmetry sector, thus improving the wavefunction
- In Monte Carlo sampling, we can project certain symmetries trivially by choosing the walkers appropriately. For example, if the walkers are \(S_z\) eigenstates:
\langle\mathbf{n}|P_{S_z}|\psi\rangle = \langle\mathbf{n}|\psi\rangle
- Breaking symmetries may lower energy, but resulting wavefunction doesn't have good quantum numbers.
- We can break and restore complex conjugation symmetry by using complex orbitals and taking the real part of the overlap.
\langle\mathbf{n}|P_{K}|\psi\rangle = \text{Re}(\langle\mathbf{n}|\psi\rangle)
H\( _8 \) linear chain (8e, 8o)
restoring \( S_z \) symmetry
restoring \( S_z \) and \( K \) symmetries
Antisymmetrized geminal power (AGP)
We can break the number symmetry as well!
|\psi\rangle = P_N\exp\left(\sum_{p,q}F_{p\uparrow,q\downarrow}a_{p\uparrow}^{\dagger}a_{q\downarrow}^{\dagger}\right)|0\rangle=\left(\sum_{p,q}F_{p\uparrow,q\downarrow}a_{p\uparrow}^{\dagger}a_{q\downarrow}^{\dagger}\right)^{N/2}|0\rangle
\( F_{p\uparrow,q\downarrow} \rightarrow\) amplitude for the bond between \( p \) and \( q \)
\langle\mathbf{n}|\psi\rangle = \text{det}(\qquad)
BCS wavefunction in real space
Spin eigenstate if \( F_{p\uparrow,q\downarrow} \) symmetric, breaks \( S^2 \) symmetry otherwise
Pfaffians
Includes same spin pairing: breaks all symmetries
|\psi\rangle =\left(\sum_{p\sigma,q\gamma}F_{p\sigma,q\gamma}a_{p\sigma}^{\dagger}a_{q\gamma}^{\dagger}\right)^{N/2}|0\rangle
\langle\mathbf{n}|\psi\rangle = \text{pfaffian}(\qquad) = \pm\left[\text{det}(\qquad)\right]^{1/2}
antisymmetric matrix
The most general mean field wavefunction: includes all others as special cases
Jastrow factor
\hat{\mathcal{J}}|\psi\rangle = \exp\left(\sum_{p\sigma,q\gamma} v_{p\sigma,q\gamma}\hat{n}_{p\sigma}\hat{n}_{q\gamma}\right)|\psi\rangle
counts site occupations and suppresses spurious ionic configurations (double occupations)
also correlates long range excitations: important for describing insulators
Wavefunction hierarchy
|\psi\rangle=\hat{\mathcal{J}}\hat{P}|\phi\rangle
jastrow
projector
reference
General form:
N\( _2 \) (14e, 18o)
| d (Bohr) | Exact | Jastrow-KSPfafian | Green's function MC |
| 1.6 | -0.5344 | -0.5337 | -0.5342 |
| 1.8 | -0.5408 | -0.5400 | -0.5406 |
| 2.5 | -0.5187 | -0.5180 | -0.5185 |
H\( _{50} \) linear chain (50e, 50o)
| U | Accurate result | Jastrow- KSGHF |
Green's function MC |
| 2 | -1.1962 | -1.1920 | -1.1939 |
| 4 | -0.8620 | -0.8566 | -0.8598 |
| 8 | -0.5237 | -0.5183 | -0.5221 |
2D Hubbard: 98 sites
Thank you!
super_group1
By Ankit Mahajan
