Basis sets unitary transformation

The Hamiltonian in Eq. [26] is usually referred to as the diabatic representation, employing the diabatic basis set <1), Hamiltonian matrix is not diagonal. There is, of course, no unique diabatic basis as any pair obtained from (]), by a unitary transformation can define a new basis. A unitary transformation defines a linear combination of cj) and < >b which, for a two-state system, can be represented as a rotation of the (]), basis on the angle /... 

The branching multiplicity index is normally ctf > 1 since T, occurs more than once the equivalent states need to be distinguished by the index a. (Only in the special case of the d2 configuration and cubic symmetry does = 1.) The appropriate basis set kets transforming according to irreducible representation r, component y, of group G are then given by a unitary transformation... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

Transformations that take one orthonormal set of basis vectors into another orthonormal set are called unitary transformations, the operators associated with them are called unitary operators. This definition preserves the norms and scalar products of vectors in Ln. The transformation (4) is in fact a set of linear equations... 

The choice of representation is arbitrary and one basis can be mapped into another by unitary transformation. Thus, let ip(n, q) and ip l, q) be two countable sets of basis vectors, such that... 

Except for the initial AO —NAO transformation, which starts from non-orthogonal AOs, each step in (3.38) is a unitary transformation from one complete orthonormal set to another. Each localized set gives an exact matrix representation of any property or function that can be described by the original AO basis. 

While all three matrices are interconvertible, the nonnegativity of the eigenvalues of one matrix does not imply the nonnegativity of the eigenvalues of the other matrices, and hence the restrictions Q>0 and > 0 provide two important 7/-representability conditions in addition to > 0. These conditions physically restrict the probability distributions for two particles, two holes, and one particle and one hole to be nonnegative with respect to all unitary transformations of the two-particle basis set. Collectively, the three restrictions are known as the 2-positivity conditions [17]. 

Physically, the 3-positivity conditions restrict the probability distributions for three particles, two particles and one hole, one particle and two holes, and three holes to be nonnegative with respect to all unitary transformations of the one-particle basis set. These conditions have been examined in variational 2-RDM calculations on spin systems in the work of Erdahl and Jin [16], Mazziotti and Erdahl [17], and Hammond and Mazziotti [33], where they give highly accurate energies and 2-RDMs. 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

The essential concept in the definition of the CDF is the use of time-dependent basis states in place of stationary basis states in the representation of the time evolution of a system, with the constraint that both sets of states are orthonormal. Consider a complete set of orthonormal stationary states S and a complete set of orthonormal time-dependent basis states D t) related by the unitary transformation U t) ... 

To construct the Fock matrix, one must already know the molecular orbitals ( ) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent field (SCF). In practice, solution of the Fock equations proceeds as follows. First transform the basis set / into an orthonormal set 2 by means of a unitary transformation (a rotation in n dimensions),... 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

There is no need to pass from the basis set of the ATs L,Mi,S,Ms) to the basis set of the atomic multiplets (LS),J,Mj) since such a unitary transformation does not lead to a gain in the computational effort. In the basis set of the L,Mi, S, Ms) functions, the operator Vax is diagonal but the operator Hso has off-diagonal matrix elements. In contrast, in the basis set of the (LS),J,Mj) kets, the operator Hso is diagonal, but the operator Vax has off-diagonal matrix elements. Therefore, none of these basis sets is appropriate for considering the Zeeman term as a small perturbation. 

Within the basis sets of a reaction, the operator Q L (or the superoperator Qkl) represented by a unit matrix. In an arbitrary orthonormal basis le,KL> in the space HKL, the operator is represented by the inverse of the unitary matrix 0 of the basis set transformation ... 

A unitary transformation modifies the eigenvectors of a hamiltonian without changing its spectrum of eigenvalues. Thus, it is possible to eliminate certain terms in H by changing the basis set, and sometimes even diagonalize H completely. To apply a unitary transformation, it is possible either to transform the entire hamiltonian, or to express the initial hamiltonian as a function of state vectors or of transformed operators according to the relations... 

There is an additional, more fundamental, issue involved in applying the standard diabatic formalism. The solvent reorganization energy and the solvent component of the equilibrium free energy gap are bilinear forms of A ab and (Fav (Eqs. [45] and [47]). A unitary transformation of the diabatic basis (Eq. [27]), which should not affect any physical observables, then changes A b and v, affecting the reorganization parameters. The activation parameters of ET consequently depend on transformations of the basis set ... 

It is stressed again that, assuming a given basis set of Is orbitals for the H atoms and 2s and 2p orbitals for C, exactly the same wavefunction is obtained irrespective of using the canonical m.o.s V i (as linear combinations of a.o.s) or the molecular orbitals Xi (linear combinations of through a unitary transformation). Hence, for example the calculated molecular energy is exactly the same, provided that the residual interactions given by... 

This matrix is nnitary, b( canse all the matrices on the right hand side of E(p(115) are unitary. It should be noted that E(i.(115) does not describe the formal transformation from the body-fixed to space-fixed frame by coordinate transformation [1], but just gives the transformation of electronic basis sets in the asymptotic region. 

It should be noted that the SO operator is nondiagonal in the diabatic spin-orbital electronic basis which usually is employed to set up the E x E JT Hamiltonian, see (28, 33). The (usually ad hoc assumed) diagonal form of H o is obtained by the unitary transformation S which mixes spatial orbitals and spin functions of the electron. In this transformed basis, the electronic spin projection is thus no longer a good quantum number. 

Given a set of geminals y>, satisfying the SO condition [Eq. (15)] and having a basis set expansion in an orthogonal set of one-electron orbitals [Eq. (5)], it is always possible to find a unitary transformation in the one-electron space so that in the new basis each geminal is expanded in a subset of the one-electron functions the subsets having no common elements. 

Here, atomic basis function. The diagonalization of D yields a new set of noncanonical, virtual MOs that is related to the original one by a unitary transformation,... 

Natural orbitals [28-39] are defined to be a basis set in terms of which the first-order density matrix is diagonal. Thus if Cb, is a unitary transformation matrix which brings pab into a diagonal form, i.e. if Cbi is a solution to the secular equation... 

In analysing the variation of the energy functional with respect to orbital variations, we need to maintain the orthonormality of the spinor basis. It is convenient to do this by constructing a unitary transformation U = I -t- T such that the set p = 1,..., A/1 WqUqp is an improved basis. 

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