Exact Hamiltonian

In this type of empirical approximation, the form of the matrix elements Hu has been specified, but it is also possible to specify the form of the wave function and therefore define the Hamiltonian exactly. Thus, if the ground-state function is a Hartree-Fock-type, the Hamiltonian can be evaluated... 

Successive orders H(n ) can be shown to correspond to successive orders in a moment (or cumulant) expansion of the propagator, which takes one to increasing times. Truncation of the chain at a given order n (i.e., 3 + 3n modes) leads to an approximate, lower-dimensional representation of the dynamical process, which reproduces the true dynamics up to a certain time. In Ref. [51], we have demonstrated explicitly that the nth-order (3n+3 mode) truncated HEP Hamiltonian exactly reproduces the first (2n + 3)rd order moments (cumulants) of the total Hamiltonian. A related analysis is given in Ref. [73],... 

The Fock space dimensionality is finite when one deals with finite model systems. Thus, it appears that we can solve the model Hamiltonians exactly for finite systems and from the finite system properties, infer the behaviour of the system in the thermodynamic limit by suitable scaling techniques. However, the dimensionality increases as (25 4-1) for a spin-S chain and as 4 for fermions, where N is the number of sites. Thus, it is very difficult to carry out brute force numerical computations on a large system and the exact diag-onalization studies are primarily restricted to quasi-one-dimensional systems with very few sites per unit cell. 

Up to this point we have just been able to decouple a potential-free Dirac Hamiltonian exactly. In the presence of a scalar potential we are able to... 

The approximate Hamiltonian matrix Ho should be chosen so that it is a good approximation to the exaa Hamiltonian H and so that Ho — is easily inverted. Usually, Ho is taken to be a diagonal matrix with elements equal to those of H. This approximation is a reasonable one since the exact Hamiltonian is often dominated by the diagonal elements. If necessary, the diagonal approximation can be refined by calculating the Hamiltonian exactly in a small subspace that corresponds to the dominant part of the wave function. 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

Hamiltonian is Hilly treated, corrections to all mean-field results should converge to the same set of exact states. In practice, one is never able to treat all corrections to any mean-field model. Thus, it is important to seek particular mean-field potentials for which the corrections are as small and straightforward to treat as possible. 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

A further model Hamiltonian that is tailored for the treatment of non-adiabatic systems is the vibronic coupling (VC) model of Koppel et al. [65]. This provides an analytic expression for PES coupled by non-adiabatic effects, which can be fitted to ab initio calculations using only a few data points. As a result, it is a useful tool in the description of photochemical systems. It is also very useful in the development of dynamics methods, as it provides realistic global surfaces that can be used both for exact quantum wavepacket dynamics and more approximate methods. 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

An alternative form of exact nonrelativistic vibration-rotation Hamiltonian for triatomic molecules (ABC) is that used by Handy, Carter (HC), and... 

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3]... 

The exact propagator for a Hamiltonian system for any given time increment At is symplectic. As a consequence it possesses the Liouville property of preserving volume in phase space. 

Symplectic integration methods replace the t-flow (pt,H by the symplectic transformation which retains Hamiltonian features of They poses a backward error interpretation property which means that the computed solutions are solving exactly or, at worst, approximately a nearby Hamiltonian problem which means that the points computed by means of symplectic integration, lay either exactly or at worst, approximately on the true trajectories [5]. 

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