State-averaging

State averaging gives a wave function that describes the first few electronic states equally well. This is done by computing several states at once with the same orbitals. It also keeps the wave functions strictly orthogonal. This is necessary to accurately compute the transition dipole moments. 

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Careful estimates suggest that the total losses to agricultural crops from insect attacks in the United States average about 10% of production and amount to more than 14 biUion annually (1). Worldwide agricultural losses from insect attacks have been estimated as about 14% of production (2). 

Equipment and material normally amount to about 40 to 45 percent of the costs of a typical chemical plant. In general, equipment and material costs are ghtly cheaper in European countries and Japan, whereas in Mexico and Canada they are nearer the United States average. 

Perform a series of state-averaged 4,4 CAS calculations on the ground state, using the active space you have selected. Normally, a CAS calculation optimizes the orbitals and hence the wavefimction for the state of interest (as specified by the NRoot option). However, in a state-averaged CAS, orbitals are optimi. ed to provide the best description of the specified group of states as a whole that is possible with a single set of orbitals... 

Run a final state-averaged calculation at the fuUy-optimized conical intersection using the 4-31G basis set and P to predict the energies of the two states and view the configuration coefficients. (This step will not be necessary if you chose to use P for the final conical intersection optimization job you ll find the relevant output in the CAS output for the final optimization step, preceding the table giving the stationary point geometry.)... 

C=C stretch 80 C-13 chemical shifts 22, 53 C60 31,32 C60O isomers 54 carbon dioxide 120, 182 carbon monoxide 175,191 carbonyl series 84 carbonyl stretch 84, 220 in solution 244 Carmichael 136 Carpenter 152, 196 Cartesian coordinates 52, 286, 287 CASSCF keyword 228 CASSCF method 228,229,230,231, 232,233, 234,235 state-averaged 233... 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

As a eonelusion, the ealculation will be performed using a state-averaged MCSCF treatment in a well-designed active space. 

The state-averaging was done for seven A, six B, ten B2 and seven A2 states for both singlet and triplet multiplicities. 

CAS F calculations are not a universal solution to symmetry-breaking of the wave functions, and for such weak resonance problems it is far more reliable to start from state average solutions which treat on an equal footing the two configurations which interact weakly. 

The extensive calculations of Serrano-Andres et al [31] have shown a spurious valence-Rydberg mixing in the CASSCF wave functions when valence (7t,7r )and Rydberg orbitals are optimized all together in a state average calculation it was shown that these orbitals loose their diffuse character and instead tend to provide an extra correlation to valence orbitals. To avoid such interaction, the orbitals used for the Cl treatment of the electronic spectrum were obtained by a two step procedure ... 

Likewise the even part of the probability can be tested by taking the steady-state average force. Using similar linearization arguments to the earlier ones, it may be shown that... 

Figure 8 shows the r-dependent thermal conductivity for a Lennard-Jones fluid (p = 0.8, 7o = 2) [6]. The nonequilibrium Monte Carlo algorithm was used with a sufficiently small imposed temperature gradient to ensure that the simulations were in the linear regime, so that the steady-state averages were equivalent to fluctuation averages of an isolated system. 

An increasing number of household batteries are being used today. On average, a person owns about two button batteries, 10 normal (A, AA, AAA, C, D, 9V, etc.) batteries, and throws out about eight household batteries per year. About 3 billion batteries are sold annually in the United States averaging about 32 per family or 10 per person.5 9 Table 29.1 indicates the typical types of household batteries. 

Fast motions of the molecules in the liquid state average all these interactions. Chemical shifts and J values are measured as discrete averages, and the dipolar and quadrupolar interactions are averaged to zero. 

Fig. 5.2 Radial distribution curves, Pv

Fe HB(C2H2N3)3 2] has been obtained by dehydration under heating of the low-spin hexahydrate. The crystal structure for this hexahydrate has been determined at room temperature [17]. It clearly contains Fe(II) ions in the low-spin state (average Fe-N distance=1.99 A). The dehydrated derivative [Fe HB(C2H2N3)3 2] has been reported to exhibit a very abrupt spin transition between 334-345 K via variable temperature UV-vis and 57Fe Mossbauer spectroscopy studies [19]. After the publication of a preliminary magnetic study in 1994 [19], a more detailed report appeared in 1998 [20]. 

Assume several CASSCF wave functions, T,-, i= l,N, obtained in a state average calculation. The corresponding (single state) CASPT2 functions are %i,i=l,N. The functions T, + x,- are used as basis functions in a variational calculation where all terms higher than second order are neglected. The corresponding effective Hamiltonian has the elements ... 

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