Two-body creation operators

For strings containing two or more elementary operators, it is possible to constmct more than one tensor operator. We shall in Section 2.6.7 present a general method for the construction of tensor operators from strings of elementary operators. At present, we note that, by coupling the two doublet operators and it is possible to generate a singlet two-body creation 

The tensor properties of these operators are verified by substitution in (2.3.1) and (2.3.2). The defining relations (2.3.1) and (2.3.2) allow components of a given tensor operator to be scaled by a common factor. The scaling factor given here for the two-body creation operators is conventional.  

We use the term two-body rather than two-electron for the ereation operators, reserving the latter term for number-conserving rank-2 operators such as the two-electron part of the Hamiltonian. In the same manner, we sometimes refer to the elementary operators in Section 2.3.2 as the one-body ereation and armihilation operators. 

The singlet two-body creation operator (2.3.16) is symmetric in the indices p and q whereas the triplet operators are antisymmetric. For p =, the singlet two-body creation operator becomes 

---

In Section 2.3.3, we coupled two doublet creation operators and obtained in this way singlet and triplet two-body creation operators. In the same manner, we now couple a creation-operator doublet with an annihilation-operator doublet in order to generate excitation operators of singlet and triplet... 

Building on our results for the two-body creation operators in Section 2.3.3, we replace by —cigp, Oga) in (2.3.16)-(2.3.19). We thus obtain the singlet excitation operator... 

The scaling factor chosen here agrees with common usage. Note that, in contrast to the two-body creation operators of Section 2.3.3, there is no permutational symmetry in the indices p and q. Thus, both the singlet and the triplet excitation operators are nonzero for p = q. This lack of permutational symmetry is expected since an excitation from 4>p to 4>g is different from an excitation from 4>g to p. Instead, the excitation operators exhibit the symmetries... 

To gain some familiarity with the singlet excitation operator, we shall consider its effect on a simple two-electron system. By substitution and expansion, the following commutator is seen to hold for the singlet excitation operator and the two-body creation operators ... 

In the derivation (2.3.41), we used the commutator (2.3.39) and the permutational symmetry of the singlet two-body creation operator. Next, we apply the same excitation operator once more ... 

Applying to the vacuum state the two-body creation operators introduced in Section 2.3.3, we obtain two closed-shell CSFs of symmetry (the bonding and antibonding configurations)... 

Since the triplet two-body creation operators are antisymmetric in the orbital indices, the commutator vanishes for the triplet state... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

Our discussion may readily be extended from 2-positivity to p-positivity. The class of Hamiltonians in Eq. (70) may be expanded by permitting the G, operators to be sums of products of p creation and/or annihilation operators for p > 2. If the p-RDM satisfies the p-positivity conditions, then expectation values of this expanded class of Hamiltonians with respect to the p-RDM will be nonnegative, and a variational RDM method for this class will yield exact energies. Geometrically, the convex set of 2-RDMs from p-positivity conditions for p > 2 is contained within the convex set of 2-RDMs from 2-positivity conditions. In general, the p-positivity conditions imply the (7-positivity conditions, where q < p. As a function of p, experience implies that, for Hamiltonians with two-body interactions, the positivity conditions converge rapidly to a computationally sufficient set of representability conditions [17]. 

Let us make clear now the correspondence between our treatment here and Erdahl s 1978 treatment [4, Sec. 8]. Erdahl works in general Fock space and his operators conserve only the parity of the number of nuclei. He exhibits two families of operators that are polynomials in the annihilation and creation operators containing a three-body and a one-body term. Generic instances of these operators are denoted y and w. The coefficients are real, and Erdahl stresses that this is essential for his treatment. The one-body term is otherwise unrestricted, but the three-body term must satisfy conditions to guarantee that y+y or H +w does not contain a six-body term. For the first family the conditions amount to the three-body term being even under taking the adjoint, and for... 

The new quantities entering in eq.(18) are the 2n-tuple contractions like qiqj qkqi having n creation and n annihilation operators. As we shall show later (viz. in eq.(36)), they are the n-body cumulants [50]. For four operators, for example, the two-body cumulant turns out to be... 

In a CAS-SCF or a CAS-CI function V o, all the n-body density matrices with hole labels factorize into antisymmetric products of one-body densities. As a result, the two and higher-body cumulants R Z are zero when all the labels a,b,c,d- ) in R Z. are holes. The only non-vanishing four- or higher-body cumulants are those with valence labels only. The mixed n-body densities with some holes and some valences are zero unless the number and indices of hole labels in the destruction operators match with those in the creation operators. In case they match, these density matrices factorize into antisymmetric products of one-body density matrices with hole labels and the various cumulants with valence labels. 

Finally, let us emphasize that in contrast to a spin-independent case, the U(n) irreps in the U(2n) D U(n) SU(2) bases are generally changed or shifted by U(2 ) operators. Thus, one-body spin-dependent operators can change the total spin [or U(n) irrep label] by A5 = 0, 1 (or Ah = 0, 2) and two-body ones by A5 = 0, 1, 2 (or Ah = 0, 2, 4) and thus take us out of the U(n) framework. Nonetheless, as we have indicated above, the U( )-adapted creation (C ) and annihilation (C) type operators— that represent very useful tensors serving as fundamental building blocks for various U(n) tensors—are also useful in the spin-dependent U(2n) case. Indeed, since xj (X, ) are vector (contragredient vector) operators when acting on the irrep modules of U(n), their MEs in the U(n) basis are clearly related to those of (Cf) operators. In view of this fact, the MEs of one-body operators must be related to those of cfCj. The latter were carefully examined in [36] and briefly reviewed above. 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

To extract the linearly independent excitations, we shall have to use the so-called singular value decomposition of the valence density matrices generated by the creation/annihilation operators with valence-labels which axe present in the particular excitation operator in T. To illustrate this aspect, let us take an example. For any excitation operator containing the destruction of a pair of active orbitals from V o the overlap matrix of all such excited functions factorize, due to our new Wick s theorem, into antisymmetric products of one-body densities with non-valence labels and a two-particle density matrix ... 

The National Commission Report to President Obama that was written in the wake of the Deepwater Horizon disaster provides a useful overview of the creation of the Minerals Management Service (MMS) in 1982, and also of the tensions that existed from its very inception between its two functions the collection of (very substantial) revenue for the United States government, and ensuring that offshore operations were conducted safely. The Commission s report indicates that the move toward risk-based analysis that had occurred in many other nations was opposed both by industry organizations and governmental bodies in the United States in the years leading up to the Deepwater Horizon catastrophe. 

The principal reason for using the second-quantization formulation of the many-body problem is that any one-particle operator, v, and any two-particle operator, g, can be written in terms of the creation and annihilation operators, and X. ... 

---