Commutators expansion rules

The proof of the theorem affirming that J8 is a proper quantum mechanical angular momentum involves only an expansion of (Ji + J2) x (Ji + J2) with subsequent use of the commutation rules for Jj and J2, and the fact that Jj and J2 commute because they act in... 

Hence, again these ip operators do not obey canonical commutation rules due to the presence of the factor J da2Pl(a2) (which is found to be divergent in perturbation expansion of the theory). 

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04>, where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 > II 0 >) = (), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or... 

In the derivation of this expression, several manipulations are made [60]. The chain rule is applied, and the integral operator and the derivative operator in (4.17) are commuted. The order in which these operations are performed can be inverted because the Taylor series expansion is written for a fixed point in space. The third term in (4.72) occurs after the peculiar velocity is introduced. The symbol corresponds to the tensor equivalent of valid for the particular case in which the argument is a vector, both calculated in accordance with (4.20). 

The corresponding Hamiltonian operator will still be given in terms of proper expansions over bilinear forms of (boson) creation and annihilation operators. (The more complex situations including half-spin particles can be addressed as well by using fermion operators [20].) The general rule is that one introduces a set of (n -I-1) boson operators b, and b (/, y = 1,. . . , n + 1) satisfying the commutation relations... 

This result for the density matrix elements is well known from the SCF theory. The same expression can also be derived directly from Eq. (7.5), by establishing the specific commutation rules between the MO and AO operators. Substitution of the MO expansion of Eq. (7.8) into the commutators yields ... 

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