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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
Step1:  Generating Wick's theorem contractions
Preliminaries
Let's start by importing wick&d and defining a Slater determinant reference
Step2:  Wick's theorem contractions
Consider the product of operators
\begin{equation}
\hat{F} \hat{T}1 = 
\left( \sum{pq}^\mathbb{O \cup V} f_{p}^{q} { \hat{a}^{p}{q} } \right)
\left( \sum{i}^\mathbb{O} \sum_{a}^\mathbb{V} t_{a}^{i} { \hat{a}^{a}_{i} } \right)
\end{equation}
which we can create with the following code
Step3:  To apply Wick's theorem to this product, you need to create a WickTheorem object. You can compute the contraction with the contraction function.
This function takes as arguments the operator expression to contract and the minimum and maximum number of second quantized operators in the final expression (minrank/maxrank). Here we request all terms with zero to four second quantized operators
Step4:  You can get a LaTeX expression with the latex() function
Step5:  Or render the expression directly in a notebook
Step6:  Let's evaluate a more complex expression
\begin{equation}
\langle \Phi | \hat{\Lambda}1 [\hat{F}, \hat{T}_1] | \Phi \rangle
\end{equation}
where 
\begin{equation}
\hat{\Lambda}_1 =
\sum{i}^\mathbb{O} \sum_{a}^\mathbb{V} \lambda_{i}^{a} { \hat{a}^{i}_{a} } 
\end{equation}
Step7:  Example
Step8:  Next, we compute the operator $e^{-\hat{T}} \hat{H} e^{\hat{T}}$ up to the term quadratic in $\hat{T}$ and compute the fully contracted term
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Python Code:
import wicked as w
from IPython.display import display, Math, Latex
def latex(expr):
    Function to render any object that has a member latex() function
    display(Math(expr.latex()))
    
w.reset_space()
w.add_space("o", "fermion", "occupied", ['i','j','k','l','m'])
w.add_space("v", "fermion", "unoccupied", ['a','b','c','d','e','f'])    
Explanation: Generating Wick's theorem contractions
Preliminaries
Let's start by importing wick&d and defining a Slater determinant reference
End of explanation
F = w.utils.gen_op('f',1,'ov','ov')
T1 = w.op('t',['v+ o'])
FT1 = F @ T1
print(FT1)
Explanation: Wick's theorem contractions
Consider the product of operators
\begin{equation}
\hat{F} \hat{T}1 = 
\left( \sum{pq}^\mathbb{O \cup V} f_{p}^{q} { \hat{a}^{p}{q} } \right)
\left( \sum{i}^\mathbb{O} \sum_{a}^\mathbb{V} t_{a}^{i} { \hat{a}^{a}_{i} } \right)
\end{equation}
which we can create with the following code:
End of explanation
wt = w.WickTheorem()
expr = wt.contract(FT1,minrank=0,maxrank=4)
expr
Explanation: To apply Wick's theorem to this product, you need to create a WickTheorem object. You can compute the contraction with the contraction function.
This function takes as arguments the operator expression to contract and the minimum and maximum number of second quantized operators in the final expression (minrank/maxrank). Here we request all terms with zero to four second quantized operators:
End of explanation
print(expr.latex())
Explanation: You can get a LaTeX expression with the latex() function
End of explanation
latex(expr)
Explanation: Or render the expression directly in a notebook
End of explanation
L1 = w.op('lambda',['o+ v'])
opexpr = L1 @ w.commutator(F,T1)
expr = wt.contract(opexpr,0,0)
latex(expr)
Explanation: Let's evaluate a more complex expression
\begin{equation}
\langle \Phi | \hat{\Lambda}1 [\hat{F}, \hat{T}_1] | \Phi \rangle
\end{equation}
where 
\begin{equation}
\hat{\Lambda}_1 =
\sum{i}^\mathbb{O} \sum_{a}^\mathbb{V} \lambda_{i}^{a} { \hat{a}^{i}_{a} } 
\end{equation}
End of explanation
E0 = w.op("E_0",[""])
F = w.utils.gen_op('f',1,'ov','ov')
V = w.utils.gen_op('v',2,'ov','ov')
H = E0 + F + V
T = w.op("t",["v+ o", "v+ v+ o o", "v+ v+ v+ o o o"])
Explanation: Example: Coupled cluster energy expression
Here we compute the coupled cluster energy expression
\begin{equation}
E = \langle \Phi | e^{-\hat{T}} \hat{H} e^{\hat{T}} | \Phi \rangle
= E_0 + \sum_{i}^\mathbb{O} \sum_{a}^\mathbb{V} f^{a}{i} t^{i}{a} + 
\frac{1}{4} \sum_{ij}^\mathbb{O} \sum_{ab}^\mathbb{V}
 (t^{i j}{a b} + 2 t^{i}{a} t^{j}{b}) v^{a b}{i j} 
\end{equation}
We first define the full Hamiltonian operator and the cluster operator
End of explanation
Hbar = w.bch_series(H,T,2)
expr = wt.contract(Hbar,0,0)
latex(expr)
Explanation: Next, we compute the operator $e^{-\hat{T}} \hat{H} e^{\hat{T}}$ up to the term quadratic in $\hat{T}$ and compute the fully contracted term
End of explanation
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