Matrix product states

Keywords strongly correlated electrons nondynamic correlation density matrix renormalization group post Hartree-Fock methods many-body basis matrix product states complete active space self-consistent field electron correlation... 

This approximation is, in essence, the DMRG ansatz for M states (More precisely, it is the ansatz used in the one-site DMRG algorithm). Note that by increasing the dimension M, we can make the ansatz arbitrarily exact. Because the wave function coefficients are obtained as a series of matrix products, the ansatz is also referred to in the literature as the matrix product state [6-9]. Combining the above ansatz for the coefficient tensor explicitly with the Slater determinants yields the full DMRG wave function,... 

The DMRG algorithm optimizes a special class of quantum states called matrix product states (MPS) [88-90]. Usually, an MPS wave function is expressed in the canonical way, employing left- or right-normalized matrices [90]. In the case of DMRG, a mixed-canonical MPS is obtained, which incorporates both left- and right-normalized matrices A,... 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

An important theorem of matrix algebra, called singular value decomposition (SVD), states that any nxp table X can be written as the matrix product of three terms U, A and V ... 

The energy Ea is a quantum term associated with the proton reaction coordinate coupling to the Q vibration, Ea = h1 /2m. and Co is the tunneling matrix element for the transfer from the 0th vibrational level in the reactant state to the 0th vibrational level in the product state. The term AQe is the shift in the oscillator equilibrium position and F L(Eq, Ea, Laguerre polynomial. For a thorough discussion of Eq. (8), see [13],... 

In the above expression, vR and Vp are the frequencies associated with the rath level of reacting proton in the reactant state and the mth level in the product state, and q is the frequency associated with the low-frequency mode developed in the BH model. The term AQ reflects the change in equilibrium distance between the reactant and product states and Cmpm(Q) is the tunneling matrix element from the nth level in the reactant state to the mth level in the product state. An explicit evaluation of the tunneling matrix element Cmpm(Q) is obtained within the WKB semiclassical framework and is given by... 

Fig. 3. McConnell s idea of the effective 2x2 Hamiltonian is based on matching the two lowest eigenvalues of the n x n state space with an effective 2 x 2 matrix. Note that the true nxn Hamiltonian can have zero direct off-diagonal coupling between reactant and product, but non-zero values in the shaded off-diagonal region. The reactant and product states may be considered trap states embedded in the higher energy medium, and the resulting lowest eigenvalues (must) correspond principally to linear combinations of the trap states...

The empirical valence bond (EVB) approach introduced by Warshel and co-workers is an effective way to incorporate environmental effects on breaking and making of chemical bonds in solution. It is based on parame-terizations of empirical interactions between reactant states, product states, and, where appropriate, a number of intermediate states. The interaction parameters, corresponding to off-diagonal matrix elements of the classical Hamiltonian, are calibrated by ab initio potential energy surfaces in solu-fion and relevant experimental data. This procedure significantly reduces the computational expenses of molecular level calculations in comparison to direct ab initio calculations. The EVB approach thus provides a powerful avenue for studying chemical reactions and proton transfer events in complex media, with a multitude of applications in catalysis, biochemistry, and PEMs. 

As we have written it in Equation (8), the DMRG wave function contains redundant variational parameters. This means that the set of variational tensors f/ni... ij/Hk in the DMRG wave function is not unique, because we can find another set of tensors whose matrix product yields an identical state. This redundancy is analogous to the redundancy of the orbital parametrization of the Hartree-Fock determinant. In the case of the DMRG wave function, we can insert a matrix T and its inverse between any two variational tensors and leave the state invariant... 

Key topics covered in the review are the analysis of the wavepacket in the exit channel to yield product quantum state distributions, photofragmentation T matrix elements, state-to-state S matrices, and the real wavepacket method, which we have applied only to reactive scattering calculations. 

The study of proton transfer in solution with coupling to a rate promoting vibration in the sense we discussed above, was pioneered by Borgis and Hanes. They used a Marcus-like model with the important addition that the tunneling matrix element between the reactant and product states is written as... 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

Inserting resolutions of the identity written in terms of tensor product states RjAajA) (with 0 < j < [n — 1)) in the coordinate and diabatic state representation, matrix elements of the time dependent density operator are conveniently written as... 

Estimates of the electronic energy of the complex system employed in the expressions eqs. (1.254), (1.256) for its PES can be further improved. For this let us notice that the solutions of the self consistent system eq. (1.246) are used as multipliers in the basis functions eq. (1.216) of the subspace Im/. It turns out that the effective Hamiltonian Heff eq. (1.232) has nonvanishing matrix elements between the ground state of eq. (1.246) and the basis product states of the subspace ImP, differing from it by two multipliers simultaneously by the wave function for the R-system and by that for the M-system ( A p. p f 0). Indeed ... 

The direct (Kronecker) product of the SU (2) matrices representing the q- and p-pararotations acts in this space with the notion that the q-dependent matrix eq. (3.48) acts on the states of the first particle and the p-dependent one on the states of the second particle in the product state. Then we form linear combinations of the above states, which correspond to specific values of the total spin and desired spatial symmetry. The combination which corresponds to the zero total spin of two particles transforms as a scalar i.e. (singlet) s-function. Those which correspond to the total spin equal to unity form the basis in the three-dimensional (triplet) space of />functions. The coordinate (x-, y-, and z-) functions are obtained as the following combinations of the states with the definite spin projections (the above product states) ... 

From a mechanistic point of view, it is useful to divide ET into two classes which differ in the magnitude of the electronic coupling matrix element, Vel. Those ET processes for which Vel > 200 cm-1 (2.4 kJ/mol) are called adiabatic because they take place solely on just one potential energy surface, namely that which connects the reactant and product states (i.e., see the dashed arrow in Fig. 5a). In this case, the large majority of reaction trajectories that reach the avoided crossing region will... 

The matrix element can be inferred from the deviations of observed properties from those expected of the reacting system in the diabatic limit. The size of those deviations tends to increase as a function of the fraction of an electron that is delocalized between donor and acceptor, where is the vertical energy difference between the diabatic reactants and products states H -p/E = app is the coefficient for mixing and V p. 

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