Here’s a Python script for solving MP2 (second-order Møller-Plesset perturbation theory) equations for the energy of a closed-shell system.

## Python script for solving mp2 equations Code 1

There are two Codes here:

### Code 1:

```
import numpy as np
def mp2_energy(t, f, v):
n = f.shape[0] // 2
e = np.zeros((n,n,n,n))
for i in range(n):
for j in range(n):
for a in range(n, 2*n):
for b in range(n, 2*n):
e[i,j,a-n,b-n] = v[i,j,a,b] - 0.5*v[i,b,a,j]
mp2 = 0.0
for i in range(n):
for j in range(n):
for a in range(n, 2*n):
for b in range(n, 2*n):
mp2 += t[i,j,a-n,b-n] * e[i,j,a-n,b-n] / (f[i,i]+f[j,j]-f[a,a]-f[b,b])
return mp2
# Example usage:
# Set up a hypothetical closed-shell system with 4 electrons and 4 basis functions
n = 4
f = np.random.rand(2*n, 2*n)
v = np.random.rand(n,n,n,n)
t = np.random.rand(n,n,n,n)
mp2 = mp2_energy(t, f, v)
print("MP2 energy: ", mp2)
```

### Explanation

This script takes as input the t amplitudes, fock matrix (f), and two-electron integrals (v), and returns the MP2 energy for the system. The implementation assumes a closed-shell system with an equal number of alpha and beta electrons.

### Formula

**The MP2 energy is computed using the formula:**

**E_MP2 = (1/4) * sum_ijab(t_ijab * (v_ijab – 0.5*v_ibaj) / (e_ijab))**

where **t** is the t-amplitude matrix, **v** is the two-electron integral tensor, **e** is the energy denominator, and **i,j,a,b** are indices over occupied and virtual orbitals.

Note that this implementation assumes that the integrals are already in the physicist’s notation **(i.e., (ij|ab))**. If your integrals are in chemist’s notation **((ij|ba))**, you will need to transform them before passing them to this function.

## Another Python Script for Solving mp2 equations

### Code 2:

```
import numpy as np
from scipy.linalg import eigvalsh
def mp2_energy(f, g, nocc, nvirt):
"""
Compute the MP2 energy of a molecular system.
Parameters
----------
f : ndarray
One-electron Hamiltonian matrix.
g : ndarray
Two-electron integrals.
nocc : int
Number of occupied orbitals.
nvirt : int
Number of virtual orbitals.
Returns
-------
mp2_energy : float
MP2 energy.
"""
# Compute the orbital energies.
e = eigvalsh(f)
# Compute the antisymmetrized two-electron integrals.
g_iajb = g - g.transpose((0, 1, 3, 2))
# Compute the MP2 amplitudes.
t = np.zeros((nocc, nocc, nvirt, nvirt))
for i in range(nocc):
for j in range(nocc):
for a in range(nvirt):
for b in range(nvirt):
t[i, j, a, b] = g_iajb[i, j, a, b] / (e[i] + e[j] - e[nocc+a] - e[nocc+b])
# Compute the MP2 energy.
mp2_energy = 0.0
for i in range(nocc):
for j in range(nocc):
for a in range(nvirt):
for b in range(nvirt):
mp2_energy += g_iajb[i, j, a, b] * (2.0*t[i, j, a, b] - t[j, i, a, b])
return mp2_energy
```

### Explanation

This script takes as input the one-electron Hamiltonian matrix `f`

, the two-electron integrals `g`

, the number of occupied orbitals `nocc`

, and the number of virtual orbitals `nvirt`

. It computes the MP2 energy using the formula:

E(mp2) = Σ(i,j,a,b) g_iajb * (2t_ijab – t_ijba)

where `g_iajb`

are the antisymmetrized two-electron integrals and `t_ijab`

are the MP2 amplitudes, given by:

t_ijab = g_iajb / (e_i + e_j – e_a – e_b)

where `e_i`

is the energy of orbital `i`

.

## Conclusion

The implementation of MP2 equations involves a significant amount of computational and mathematical complexities. Therefore, I recommend consulting the literature or seeking help from a qualified expert in the field of quantum chemistry for guidance on how to implement the equations.

In conclusion, solving MP2 equations requires a deep understanding of the underlying principles of quantum chemistry and the necessary computational techniques. Therefore, I recommend seeking guidance from an expert in the field.

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