Off-diagonal blocks

Note Because of its neglect of off-diagonal blocks, this optimizer can sometimes oscillate and fail to converge. In this case, use a conjugate gradicn t method. 

Once the density matrix has been transformed to the NAO basis, bonds between atoms may be identified from the off-diagonal blocks. The procedure involves the following steps. 

Since S is a symmetric matrix equal to Q(0), these equalities show that the off-diagonal blocks must vanish at x = 0, and hence that there is no instantaneous coupling between variables of opposite parity. The symmetry or asymmetry of the block matrices in the grouped representation is a convenient way of visualizing the parity results that follow. 

In this case, X is to be determined by requiring that the off-diagonal blocks of the resulting transformed Dirac Hamiltonian vanishes. It can be shown that the equation for X is identical to the one we obtained in the case of a unitary transformation as given in equation (38). In this case, the effective Hamiltonian hn and wave function xp, can be written as... 

In our first exploration of the T1 and T2 conditions [5] we obtained results of the RDM method for the ground-state energy and dipole moment for a collection of small molecules and molecular ions, both closed-shell and open-shell systems. (We don t mean closed shell in a strict sense, and we only constrained the spin and spin multiplicity eigenvalues, not the elements of the RDM.) The choice of molecules and configurations largely followed Ref. [18]—a paper that, we think, reinvigorated the classical RDM approach. We showed that the addition of the T1 and T2 conditions (T2 without the off-diagonal block X)... 

We define diagonal block matrix a matrix X whose off-diagonal blocks (Xhk with hvi k) are null. In this case we prefer, for simplicity, to denote the diagonal blocks by Xk instead of Xkk. Obviously if X is a rectangular matrix, the Xk blocks are also rectangular. It is the case of the T matrix. 

The new functions (x p have the characteristic that they evolve into the original functions as the "coupling", as represented in the Sp>v matrix s off-diagonal blocks, disappears. 

The Breit interaction matrix can be treated in a similar way. The off-diagonal blocks can be written in terms of the magnetic fields using... 

Off-diagonal block of the orbital Hessian matrix used in TDDFT... 

Computationally, this can be implemented by discarding all off-diagonal blocks of the original 1-matrix p, and averaging the diagonal blocks that belong to different rows of the same irrep. It should be noted that this implementation assumes that the phase relationships between partner functions of a given irrep are the same for all MOs in the irrep, and care should be taken to ensure this is the case in practice. 

Then let us take into account the form of the matrices V. It represents an off-diagonal matrix block having nonvanishing matrix elements only if one of the vectors (bra) belongs to the subset of the occupied MOs and another (ket) to the subset of the vacant MOs. Then the only relevant part of the perturbation matrix W is the sum of two similar conjugate off-diagonal blocks ... 

The general equations for iteratively finding the off-diagonal blocks of elements of the U matrix are easily developed and implemented. Letting a collectively represent differentiation to the ith, jth, and so forth powers of the parameters,... 

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