Closed shell determinant

If, in addition, singly excited states with respect to ri 0 are included, it can be shown (Brillouin theorem >) that the electronic ground state will still be described by the single closed-shell determinant wave function zlg of energy q. 

The size ofthe MR space is given G stands for the closed shell determinant. nS stands forn single excitations (two spin partners each). 

For the analysis of the various formalisms, manipulation of the equations, generating normal product of terms via Wick s theorem, and particularly for indicating how the proofs of the several different linked cluster theorems are achieved, we shall make frequent use of diagrams. For the sake of uniformity, we shall mostly adhere to the Hugenholtz convention/1/. All the constituents of the diagrams will be operators in normal order with respect to suitable closed-shell determinant taken as the vacuum. We shall refer to the creation/annihilation operators with respect to this vacuum after the h-p transformation.The hamiltonian H will also be taken to be in normal order with respect to... 

Consider a trial Kohn-Sham (KS) determinant, either a closed shell determinant or an open-shell high-spin determinant where all singly occupied orbitals have a spin. It is parameterized by a real unitary exponential operator, and the purpose of the transformation is to transform the orbitals to a state of minimum energy... 

In the following we divide the internal orbital space into three subspaces the core orbitals, which are doubly occupied and not correlated the closed-shell orbitals, which are correlated and the active orbitals, which are only partially filled in the MCSCF wavefunction. We will show that it is sufficient to evaluate coupling coefficients only for the active subspace. For the special case that the reference wavefunction is a single closed-shell determinant, the algorithm then reduces to the closed-shell SCEP method described by Meyer, and no coupling coefficients have to be calculated explicitly. ... 

The Coulson-Fischer wave function for H2 can be considered as the start of the Unrestricted Hartree-Fock (UHF) approach in quantum chemistry, which is the most general single determinant method. We shall not proceed further along this line, but instead ask ourselves if there is a way to correct the simation such that we obtain a wave function that dissociates correctly while preserving the spin and space symmetry of the wave function. The CF wave function gives acmally a hint. What happens if we simply skip the trouble-some triplet term in Eq. (22). This gives rise to a wave function that is a linear combination of two closed shell determinants ... 

The coupling coefficients for a particular choice of Cl expansion are obviously fixed by the expressions for the Cl matrix elements. For example the expressions in Appendix 20.B contain all the information to generate the coupling coefficients for Doubly-excited Cl for singlet states generated from a closed-shell determinant. Equally obviously, the expressions of Slater s Rules in Appendix 2. A imply the coupling coefficients for Cl using determinants. 

In practice, there is quite a difference between atoms and molecules as far as the representation of the one-particle spectrum is concerned. In atomic physics, one often utilizes a radial-angular representation of the one-electron orbitals in order to allow for the analytic integration over all spin-angular coordinates of the system [35,36]. This so-called angular reduction will be briefly discussed below in Subsection 4.4. However, in order to exploit the symmetry of free atoms the reference state must coincide with a closed-shell determinant o> = < >>, i.e. a reference state which should not depend on the magnetic quantum numbers of the one-electron functions. Unfortunately, the complexity of the perturbation expansions increases very rapidly if the number of electrons in the physical states of interest differ from (the number of electrons in) the reference state. 

All the electron spins are paired in a closed-shell determinant, and it is not surprising that a closed-shell determinant is a pure singlet. That is, it is an eigenfunction of with eigenvalue zero,... 

The closed-shell Hartree-Fock energy was derived in Subsection 2.3i as an example of the transition from spin orbitab to spatial orbitals. For the closed-shell determinant, o) l i i w/2i N/2>. ... 

Here, is the singly excited triplet configuration defined in Eq. (2.261). The closed-shell determinants and 2i 2> of course, singlets. 

The latter corresponds to a two-configuration Cl wave function analogous to Eq. 3.7, be it with fixed coefficients Ci = -C2 = 1 /V2. Note that it is tempting to characterize 3.12a as an open shell wave function, while the same wave function in 3.12b takes the form of a superposition of two closed shell determinants. The simplest VB representation for the A/ = 0 component of the lowest spin-triplet state is... 

We limit discussion to the case in which i/r is a single, closed-shell determinant. We will develop our arguments by referring to a four-electron example ... 

If the ground state can be represented by a restricted closed shell determinant, the density difference between state I and the ground state is given by (after integrating over the electron coordinates)... 

If the reference is a closed-shell determinant, it is symmetric under time reversal. We require that the coupled-cluster wave function is also symmetric under time reversal, because the wave function is nondegenerate and has an even number of electrons. Then... 

For large systems, we cannot diagonalize the density matrix to find out whether it represents a closed-shell determinant. We shall therefore express the conditions on the density matrix in an alternative form, which does not require its diagonalization. Introducing... 

Whereas the symmetry condition (10.7.17) and the trace condition (10.7.18) are shared by all fV-electron wave functions, the idempotency condition (10.7.19) is a special property of the scaled density matrix of a closed-shell determinant of doubly occupied orbitals, arising since the eigenvalues of such matrices do not change when they are squared. In the next subsection, we shall consider the corresponding conditions on the AO density matrix, in terms of which the optimization of the Hartree-Fock state will be carried out. 

The scaled AO density matrix of a closed-shell determinant R therefore satisfies the following three conditions... 

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