Notes on Antisymmetrized Product of Rank-Two Geminals
Antisymmetrized Product of Rank-Two Geminals (APr2G)
Mathematical Notes and Intuition
This note summarizes the key mathematical ideas behind the antisymmetrized product of rank-two geminals (APr2G) introduced by Johnson et al. in the paper A size-consistent approach to strongly correlated systems using a generalized antisymmetrized product of nonorthogonal geminals.
Goals:
- Understand why permanents appear in APIG.
- See why they are computationally hard.
- Understand how Borchardt’s theorem converts permanents into determinants.
- See how this leads to the APr2G ansatz.
- Clarify connections to AGP, APSG, Richardson states, and projected HFB.
1. APIG Wavefunction
The antisymmetrized product of interacting geminals (APIG) is
\[| \Psi \rangle = \prod_{p=1}^{P} \left( \sum_{i=1}^{K} c_{i,p} \ a_i^\dagger a_{\bar{i}}^\dagger \right) |0\rangle\]where
- $P$ = number of electron pairs
- $K$ = number of spatial orbital pairs
- $c_{i,p}$ = geminal coefficients
- $a_i^\dagger a_{\bar{i}}^\dagger$ creates a singlet pair in orbital $i$
Because electrons are fermions,
\[(a_i^\dagger a_{\bar{i}}^\dagger)^2 = 0\]so each orbital pair can be occupied at most once. Therefore,
\[P \le K\]2. Expansion into Slater Determinants
Choose a Slater determinant where orbital pairs
\[\{ i_1, i_2, \dots, i_P \}\]are occupied.
Each geminal must choose one of these orbitals.
There are $P!$ possible assignments:
- geminal 1 chooses $i_{\sigma(1)}$
- geminal 2 chooses $i_{\sigma(2)}$
- …
- geminal $P$ chooses $i_{\sigma(P)}$
where $\sigma$ is a permutation.
The determinant coefficient becomes
\[\sum_{\sigma} \prod_{p=1}^{P} c_{i_{\sigma(p)},p}\]This is exactly the permanent of a matrix.
3. Why the Matrix is P x P
For a fixed determinant:
- Rows correspond to geminals $p = 1, \dots, P$
- Columns correspond to the $P$ occupied orbital pairs
The coefficient matrix has dimension
\[P \times P\]The Slater determinant coefficient is
\[\mathrm{perm}(C)\]4. Why Permanent (Not Determinant)
Pair creation operators commute:
\[\hat g_i^\dagger \hat g_j^\dagger = \hat g_j^\dagger \hat g_i^\dagger\]Swapping two geminals does not introduce a minus sign.
Therefore we get a permanent, not a determinant.
5. Computational Difficulty
Permanent evaluation scales as
\[\mathcal{O}(P!)\]This is #P-hard.
General APIG is therefore computationally intractable.
6. The Key Idea: Cauchy Structure
Johnson et al. impose the special form
\[c_{i,p} = \frac{1}{e_i - k_p}\]This is a Cauchy matrix.
For a Cauchy matrix $C$, Borchardt’s theorem states
\[\mathrm{perm}(C) = \frac{\det(C \circ C)}{\det(C)}\]where
- $C \circ C$ is the elementwise square
- determinants are polynomial-time computable
Thus,
Permanent -> ratio of determinants
This reduces exponential scaling to polynomial scaling.
7. Pfaffians and Hafnians
The discussion connects to broader pairing algebra.
Determinant:
- Appears with antisymmetry (fermions)
- Used in Slater determinants
Permanent:
- Appears with commuting objects
- Appears in APIG
Pfaffian:
- Defined for antisymmetric matrices
- Satisfies
- Appears in BCS and Pfaffian QMC wavefunctions
Hafnian:
- Bosonic analogue of Pfaffian
- Counts pairings without antisymmetry
- Appears in bosonic Gaussian states
Conceptually:
Determinant -> fermions
Permanent -> boson-like pairing
Pfaffian -> fermionic pairing
Hafnian -> bosonic pairing
APr2G sits inside this broader pairing structure.
8. Final APr2G Wavefunction
With the Cauchy parametrization:
\[|\Psi_{\mathrm{APr2G}}\rangle = \prod_{p=1}^{P} \left( \sum_{i=1}^{K} \frac{1}{e_i - k_p} a_i^\dagger a_{\bar{i}}^\dagger \right) |0\rangle\]The Slater determinant coefficients become
\[\phi_{\{i\}} = \frac{\det(C \circ C)}{\det(C)}\]This makes the ansatz polynomially tractable.
9. Hierarchy and Connections
APr1G = AGP
If
\[c_{i,p} = f_i\](all geminals identical), we obtain the antisymmetrized geminal power (AGP).
AGP -> Slater determinant
If only $P$ orbitals have nonzero $f_i$, AGP reduces to a single Slater determinant.
APr2G -> APSG
If parameters separate into orbital blocks, geminals become strongly orthogonal and we recover APSG.
APr2G -> Richardson wavefunction
If $k_p$ satisfy Richardson equations, APr2G reproduces exact pairing Hamiltonian eigenstates.
AGP <-> Projected HFB
AGP is equivalent to number-projected Hartree-Fock-Bogoliubov theory.
Hierarchy:
Slater $\subset$ AGP $\subset$ APr2G
10. Beauty of This Work
- Makes nonorthogonal geminals computationally feasible.
- Uses Borchardt’s theorem to bypass permanent scaling.
- Unifies AGP, APSG, Richardson, and projected HFB.
- Provides size-consistent polynomial scaling.
- Connects quantum chemistry and integrable pairing models.