Operator expansion

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

A computer algorithm has been developed for making multi-component mixture calculations to predict (a) thermodynamic properties of liquid and vapor phases (b) bubble point, dew point, and flash conditions (c) multiple flashes, condensations, compression, and expansion operations and (d) separations by distillation and absorption. 

Figure 3. Combining preparation of weekly bulletin with fie expansion operations...

Combining Preparation of Weekly Bulletin with File Expansion Operations... 

From 1963 to 1967 a 1/32 inch extrudate catalyst was used. In 1967 relatively minor modifications were made to accommodate a micro-spheroidal fine catalyst. This eliminated the need for the internal recycle pump previously required to supply the liquid velocity necessary for bed expansion. Operating and performance data have been described previously (3, 4). 

The Viterbi algorithm is based on trellis expansion operation. During the execution of Viterbi algorithm implemented in assembly language programming, code for trellis expansion function is executed multiple times. For example, if there are 12 bits in a received codeword then this trellis function is called approximately nineteen times, as shown in Fig. 4.2. 

Theorem 6.16. A generalized simplicial complex A is contractible if and only if there exists a sequence of collapses and expansions (operation inverse to the collapse, also called an anticollapse) leading from A to a vertex. 

He is able to generalize this to include complex reversible cycles - we have come a long way from the simple expansive operation of steam-engines - and to lay down that in such complex cycles the quantities of heat are related to the temperatures (absolute) at which they are absorbed or transmitted by the linear equation ... 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

Now consider die case where Ais itself a time-independent operator, such as that for the position, momenPiin or angidar momenPiin of a particle or even the energy of the benzene molecule. In these cases, the time-dependent expansion coefficients are unaffected by application of the operator, and one obtains... 

We deal witii the exponentials in (equation Al.4.102) and (equation Al.4.105) whose arguments are operators by using their Taylor expansion... 

It can be shown [ ] that the expansion of the exponential operators truncates exactly at the fourth power in T. As a result, the exact CC equations are quartic equations for the t y, etc amplitudes. The matrix elements... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

In order to incorporate the geometric phase effect in a formulation based on an expansion in G-H basis functions we need to consider the operation of the momentum operator on a basis function, that is, to evaluate terms as... 

However, this procedure depends on the existence of the matrix G(R) (or of any pure gauge) that predicates the expansion in Eq. (90) for a full electronic set. Operationally, this means the preselection of a full electionic set in Eq. (129). When the preselection is only to a partial, truncated electronic set, then the relaxation to the truncated nuclear set in Eq. (128) will not be complete. Instead, the now tmncated set in Eq. (128) will be subject to a YM force F. It is not our concern to fully describe the dynamics of the truncated set under a YM field, except to say (as we have already done above) that it is the expression of the residual interaction of the electronic system on the nuclear motion. 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

Each of these individual terms can be simplified if we recognise that terms dependent upon electrons other than those in the operator can be separated out. For example, the first term in the expansion, Equation (2.77), is ... 

When the Coulomb and exchange operators are expressed in terms of the basis functions and the orbital expansion is substituted for xu then their contributions to the Fock matrix element take the following form ... 

If, on the other hand, the thermometer has previously been used at some temperature below the freezing-point of benzene, when the bulb is originally placed in the beaker of water at 7-8 C., the mercury will rise in the capillary and ultimately collect in the upper part of the reservoir at a. When the expansion is complete, again tap the thermometer sharply at R so that this excess of mercury drops down into b, and then as before check the success of the setting by placing the thermometer m some partly frozen benzene. In either case, if the adjustment is not complete, repeat the operations, making a further small adjustment, until a satisfactory result is obtained. 

If F is the operator for momentum in the x-direction andA (x,t) is the wave function for x as a function of time t, then the above expansion corresponds to a Fourier transform o/ P... 

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