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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:  Algebraic expressions and the inner workings of Wick&d
The main wick&d workflow and corresponding object classes
By now you have seen how to define operators, combine them into expression, and apply Wick's theorem to bring products of operators into normal ordered form. This process is illustrated in the following picture, together with the corresponding C++ class that encode these quantities.
So far we have encountered three main objects
Step2:  Orbital indices
To identify orbitals wick&d defines a class (Index) that stores indices efficiently.
To create an index we need to specify the space and a cardinal number associated with the index (to distinguish multiple indices that refer to the same space). Here we create the index for an occupied orbital, $o_0$, by calling the index function passing the string o_0. We can alternatively omit the underscore and just call this function with o0
Step3:  Indices have two attributes, the orbital space and position
Step4:  When we render this to LaTeX using the (member) funtion latex(), wick&d uses pretty indices instead
Step5:  In this notebook we also defined a function called latex() that can render any object that has a member function called latex(). Here is what happens if we print some indices
Step6:  Second quantized operators
We can similarly construct second quantized operators for fermions and bosons using the functions cre and ann. The following is an example of creating the fermionic operators $\hat{a}{o_0}$ and $\hat{a}^\dagger{o_1}$. Note that the type of field (fermion/boson) is consistent with the one specified when defining the orbitals space information.
Step7:  Creation operators are indicated with a+, while annihilation operators with with a-. A second quantized operator has function to get its properties
Step8:  Tensors
Another basic class in wick&d is Tensor, used to represent tensors. Tensor creation follows an approach similar to that of creating SQOperators, where we have to specify the tensor indices.
Here we start by creating the tensor $T_{v_0}^{o_0}$ using the function tensor. This function take a label ("T"), a list of lower indices (specified as a product of space label and index cardinal number), a list of upper indices, and the tensor symmetry.
The allowed values for the tensor symmetry are
Step9:  We can grab properties of tensors with the following functions
Step10:  Here is a more elaborate example that builds the antisymmetric four-index tensor $V_{v_0 v_1}^{o_0 o_1}$
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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: Algebraic expressions and the inner workings of Wick&d
The main wick&d workflow and corresponding object classes
By now you have seen how to define operators, combine them into expression, and apply Wick's theorem to bring products of operators into normal ordered form. This process is illustrated in the following picture, together with the corresponding C++ class that encode these quantities.
So far we have encountered three main objects:
1. The WickTheorem class, responsible for performing Wick's theorem contractions.
1. The OperatorExpression class, responsible for representing a linear combination of operator products.
1. The Expression class, a class that represents the algebraic terms generated by contracting an operator expression.
The first two classes belong to the diagrammatic side of wick&d. This part of the code represents operators with in a compact way that is convenient when applying Wick's theorem.
The Expression class belongs instead to the algebraic side of wick&d, which is responsible for representing general algebraic expressions.
<img src="fig-05-diagram.pdf">
Hierarchy of algebraic classes
The Expression class represents the most general type of algebraic expression involving tensors and second quantized operators. This class is built from a series of basic classes that represent the following objects:
- Index: an orbital index
- Tensor: a tensor
- SQOperator: a second quantized operator
- SymbolicTerm: a product of tensors and second quantized operators
- scalar_t: a rational scalar
- Expression: a linear combination of symbolic terms
There is also the class Term to store the product of a symbolic term and a scalar. Howeverm, it is used mostly in the C++ side to pass information among classes.
The reason wick&d splits a term into a symbolic part and a scalar is because then an Expression can be represented with a map SymbolicTerm $\rightarrow$ scalar_t. The map structure facilitates the simplification of expressions.
$$
\underbrace{
    h_{p}^{q}
}{\mathrm{Tensor}}
\hat{a}_p^\dagger \underbrace{
    \hat{a}_q
}{\mathrm{SQOperator}}
$$
$$
\underbrace{
\underbrace{
    h_{p}^{q} \hat{a}p^\dagger \hat{a}_q
}{\mathrm{Term}}
+
\underbrace{
    \frac{1}{4}
\underbrace{
    v_{pq}^{rs} \hat{a}p^\dagger \hat{a}_q^\dagger \hat{a}_s \hat{a}_r
}{\mathrm{SymbolicTerm}}
}{\mathrm{Term}}
}{\mathrm{Expression}}
$$
End of explanation
idx = w.index('o_0')
idx2 = w.index('o0')
print(idx)
print(idx2)
Explanation: Orbital indices
To identify orbitals wick&d defines a class (Index) that stores indices efficiently.
To create an index we need to specify the space and a cardinal number associated with the index (to distinguish multiple indices that refer to the same space). Here we create the index for an occupied orbital, $o_0$, by calling the index function passing the string o_0. We can alternatively omit the underscore and just call this function with o0:
End of explanation
idx.pos(), idx.space()
Explanation: Indices have two attributes, the orbital space and position
End of explanation
print(idx)
idx.latex()
Explanation: When we render this to LaTeX using the (member) funtion latex(), wick&d uses pretty indices instead
End of explanation
latex(w.index('o_0'))
latex(w.index('o_2'))
latex(w.index('v_0'))
Explanation: In this notebook we also defined a function called latex() that can render any object that has a member function called latex(). Here is what happens if we print some indices
End of explanation
cre = w.cre('o0')
ann = w.ann('o1')
cre, ann
Explanation: Second quantized operators
We can similarly construct second quantized operators for fermions and bosons using the functions cre and ann. The following is an example of creating the fermionic operators $\hat{a}{o_0}$ and $\hat{a}^\dagger{o_1}$. Note that the type of field (fermion/boson) is consistent with the one specified when defining the orbitals space information.
End of explanation
ann.field_type(), ann.type(), ann.index()
Explanation: Creation operators are indicated with a+, while annihilation operators with with a-. A second quantized operator has function to get its properties
End of explanation
t = w.tensor(label="t", lower=['v0'], upper=['o0'], symmetry=w.sym.none)
latex(t)
Explanation: Tensors
Another basic class in wick&d is Tensor, used to represent tensors. Tensor creation follows an approach similar to that of creating SQOperators, where we have to specify the tensor indices.
Here we start by creating the tensor $T_{v_0}^{o_0}$ using the function tensor. This function take a label ("T"), a list of lower indices (specified as a product of space label and index cardinal number), a list of upper indices, and the tensor symmetry.
The allowed values for the tensor symmetry are:
- w.sym.none: No symmetry
- w.sym.symm: Symmetric with separate permutations of upper and lower indices
- w.sym.anti: Antisymmetric with separate permutations of upper and lower indices
End of explanation
t.label(), t.upper(), t.lower(), t.symmetry()
Explanation: We can grab properties of tensors with the following functions
End of explanation
t = w.tensor("V",['v0','v1'],['o0','o1'],w.sym.anti); latex(t)
t.label(), t.upper(), t.lower(), t.symmetry()
Explanation: Here is a more elaborate example that builds the antisymmetric four-index tensor $V_{v_0 v_1}^{o_0 o_1}$
End of explanation
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