Hamiltonian eigenstate

It is evident that a 2-RDM that corresponds to a Hamiltonian eigenstate also corresponds to a pure-spin state. However, when one is working with an approximated RDM, it is important that this RDM should correspond to a spin eigenstate. 

At the basis of the theoretical developments leading to the MCSEs lie a set of relations reported by Tel et al. [81], which link CMs of different orders among themselves. These C-relations establish a set of necessary conditions that the CMs must satisfy when they correspond to a Hamiltonian eigenstate. The... 

Consider then the case of bichromatic control where a system prepared in a. superposition of bound Hamiltonian eigenstates IE,),... 

To demonstrate control of chaotic dynamics, we assume that an initial superposition of Hamiltonian eigenstates of the form [231]... 

The above calculations provided the electronic ground and the first nine excited energies as well as the corresponding (transition) dipoles, at each point of the above reaction path. Such unperturbed Hamiltonian eigenstates defined the basis set used to construct the perturbed Hamiltonian matrix, Eq. 8-1, which was then diagonalized at each simulation frame, leading to the reaction free energy and related properties. 

The ket Qo, ) is a position eigenstate for the nuclei and a Hamiltonian eigenstate for the electrons. In the case of finite time, we can always divide it into N small time steps e and integrate over the intermediate nuclear positions ... 

In the basis of the Hamiltonian eigenstates, the thermal equilibrium density matrix constructed from Equation (2.5.7) is diagonal ... 

We now add die field back into the Hamiltonian, and examine the simplest case of a two-level system coupled to coherent, monochromatic radiation. This material is included in many textbooks (e.g. [6, 7, 8, 9, 10 and 11]). The system is described by a Hamiltonian having only two eigenstates, i and with energies = and Define coq = - co. The most general wavefunction for this system may be written as... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

Often the eigenstates of the Hamiltonian are not known. Then one uses an appropriate set of states u) which... 

Suppose that x [Q)) is the adiabatic eigenstate of the Hamiltonian H[q]Q), dependent on internal variables q (the electronic coordinates in molecular contexts), and parameterized by external coordinates Q (the nuclear coordinates). Since x Q)) must satisfy... 

Equations (16)-(20) show that the real adiabatic eigenstates are everywhere smooth and continuously differentiable functions of Q, except at degenerate points, such that E (Q) — E, [Q) = 0, where, of com se, the x ) are undefined. There is, however, no requirement that the x ) should be teal, even for a real Hamiltonian, because the solutions of Eq. fl4) contain an arbitrary Q dependent phase term, gay. Second, as we shall now see, the choice that x ) is real raises a different type of problem. Consider, for example, the model Hamiltonian in Eq. (8), with / = 0 ... 

Mead and Truhlar [10] broke new ground by showing how geometric phase effects can be systematically accommodated in scattering as well as bound state problems. The assumptions are that the adiabatic Hamiltonian is real and that there is a single isolated degeneracy hence the eigenstates n(q-, Q) of Eq. (83) may be taken in the form... 

Given a real electronic Hamiltonian, with single-valued adiabatic eigenstates of the form n) = and x ), the matrix elements of A become... 

A time-varying wave function is also obtained with a time-independent Hamiltonian by placing the system initially into a superposition of energy eigenstates ( n)), or forming a wavepacket. Frequently, a coordinate representation is used for the wave function, which then may be written as... 

Free Panicle in ID. The Hamiltonian consists only of the kinetic energy of the particle having mass m ([237] Section 28, [259]). The (unnormalized) energy eigenstates labeled by the momentum index k aie... 

Interaction with light changes the quantum state a molecule is in, and in photochemistry this is an electronic excitation. As a result, the system will no longer be in an eigenstate of the Hamiltonian and this nonstationaiy state evolves, governed by the time-dependent Schrddinger equation... 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

In most cases, this Lanczos-based technique proves to be superior to the Chebyshev method introduced above. It is the method of choice for the application problems of class 2b of Sec. 2. The Chebyshev method is superior only in the case that nearly all eigenstates of the Hamiltonian are substantially occupied. 

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

Operators that eommute with the Hamiltonian and with one another form a partieularly important elass beeause eaeh sueh operator permits eaeh of the energy eigenstates of the system to be labelled with a eorresponding quantum number. These operators are ealled symmetry operators. As will be seen later, they inelude angular momenta (e.g., L2,Lz, S, Sz, for atoms) and point group symmetries (e.g., planes and rotations about axes). Every operator that qualifies as a symmetry operator provides a quantum number with whieh the energy levels of the system ean be labeled. 

This spatial distribution is not stationary but evolves in time. So in this ease, one has a wavefunetion that is not a pure eigenstate of the Hamiltonian (one says that E is a superposition state or a non-stationary state) whose average energy remains eonstant (E=E2,i ap + El 2 bp) but whose spatial distribution ehanges with time. 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

Beeause the total Hamiltonian of a many-eleetron atom or moleeule forms a mutually eommutative set of operators with S, Sz, and A = (V l/N )Zp Sp P, the exaet eigenfunetions of H must be eigenfunetions of these operators. Being an eigenfunetion of A forees the eigenstates to be odd under all Pij. Any aeeeptable model or trial wavefunetion should be eonstrained to also be an eigenfunetion of these symmetry operators. 

It has been demonstrated that a given eleetronie eonfiguration ean yield several spaee- and spin- adapted determinental wavefunetions sueh funetions are referred to as eonfiguration state funetions (CSFs). These CSF wavefunetions are not the exaet eigenfunetions of the many-eleetron Hamiltonian, H they are simply funetions whieh possess the spaee, spin, and permutational symmetry of the exaet eigenstates. As sueh, they eomprise an aeeeptable set of funetions to use in, for example, a linear variational treatment of the true states. 

The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a so-called asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the J, M, and K quantum numbers. However, given the three principal moments of inertia la, Ib, and Ic, a matrix representation of each of the three contributions to the rotational Hamiltonian... 

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