Single determinant

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

In this model, both the parent moleeule and the speeies generated by adding or removing an eleetron are treated at the single-determinant level. 

So, within the limitations of the single-determinant, frozen-orbital model set forth, the ionization potentials (IPs) and eleetron affinities (EAs) are given as the negative of the oeeupied and virtual spin-orbital energies, respeetively. This statement is referred to as Koopmans theorem (T. Koopmans, Physiea 1, 104 (1933)) it is used extensively in quantum ehemieal ealeulations as a means for estimating IPs and EAs and often yields results that are at least qualitatively eorreet (i.e., 0.5 eV). 

CIS calculations from the semiempirical wave function can be used for computing electronic excited states. Some software packages allow Cl calculations other than CIS to be performed from the semiempirical reference space. This is a good technique for modeling compounds that are not described properly by a single-determinant wave function (see Chapter 26). Semiempirical Cl... 

The above problem with H2 dissociation is a matter of wave function construction. The functional form of a restricted single-determinant wave function will not allow a pair of electrons in an orbital to separate into two different orbitals. Wave function construction issues were addressed in greater detail in Chapters 3 through 6. 

Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination. 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

The calculation mixes all single determinant wavefunctions that can be obtained from the ground state by exciting electrons from a subset of the occupied orbitals (of the ground state) to a subset of the unoccupied orbitals. The subsets are specified as a fixed number (highest occupied or lowest unoccupied) or by an energy criterion associated with the energy difference between the occupied orbital and the unoccupied orbital. 

The UHF option allows only the lowest state of a given multiplicity to be requested. Thus, for example, you could explore the lowest Triplet excited state of benzene with the UHF option, but could not ask for calculations on an excited singlet state. This is because the UHF option in HyperChem does not allow arbitrary orbital occupations (possibly leading to an excited single determinant of different spatial symmetry than the lowest determinant of the same multiplicity), nor does it perform a Configuration Interaction (Cl) calculation that allows a multitude of states to be described. 

For some systems a single determinant (SCFcalculation) is insufficient to describe the electronic wave function. For example, square cyclobutadiene and twisted ethylene require at least two configurations to describe their ground states. To allow several configurations to be used, a multi-electron configuration interaction technique has been implemented in HyperChem. 

HyperChem uses single determinant rather than spin-adapted wave functions to form a basis set for the wave functions in a configuration interaction expansion. That is, HyperChem expands a Cl wave function, in a linear combination of single Slater determ in ants... 

Sometimes just one determination is available on each of several known materials similar in composition. A single determination by each of two procedures (or two analysts) on a series of material may be used to test for a relative bias between the two methods, as in Example 2.4. Of course, the average difference does not throw any light on which procedure has the larger constant error. It only supplies a test as to whether the two procedures are in disagreement. 

Determine the error (a = 0.05) for the following situations. In each case, assume that the variance for a single determination is 0.0025 and that the variance for collecting a single sample is 0.050. (a) Nine samples are collected, each of which is analyzed once, (b) One sample is collected and analyzed nine times, (c) Five samples are collected, each of which is analyzed three times. 

The one-point fixed-time integral method has the advantage of simplicity since only a single measurement is needed to determine the analyte s initial concentration. As with any method relying on a single determination, however, a... 

Partitioning of random error, systematic errors due to the analyst, and systematic error due to the method for (a) replicate analyses performed by a single analyst and (b) single determinations performed by several analysts. 

In the two-sample collaborative test, each analyst performs a single determination on two separate samples. The resulting data are reduced to a set of differences, D, and a set of totals, T, each characterized by a mean value and a standard deviation. Extracting values for random errors affecting precision and systematic differences between analysts is relatively straightforward for this experimental design. 

Construction of Property Control Charts The simplest form for a property control chart is a sequence of points, each of which represents a single determination of the property being monitored. To construct the control chart, it is first necessary to determine the mean value of the property and the standard deviation for its measurement. These statistical values are determined using a minimum of 7 to 15 samples (although 30 or more samples are desirable), obtained while the system is known to be under statistical control. The center line (CL) of the control chart is determined by the average of these n points... 

Determinarion of MW and MWD by SEC using commercial narrow molecular weight distribution polystyrene as calibration standards is an ASTM-D5296 standard method for polystyrene (11). However, no data on precision are included in the 1997 edition of the ASTM method. In the ASTM-D3536 method for gel-permeation chromatography from seven replicates, the M of a polystyrene is 263,000 30,000 (11.4%) for a single determination within the 95% confidence level (12). A relative standard deviation of 3.9% was reported for a cooperative determination of of polystyrene by SEC (7). In another cooperative study, a 11.3% relative standard deviation in M, of polystyrene by GPC was reported (13). 

Configuration Interaction (Cl) methods begin by noting that the exact wavefunction 4 cannot be expressed as a single determinant, as Hartree-Fock theory assumes. Cl proceeds by constructing other determinants by replacing one or more occupied orbitals within the Hartree-Fock determinant with a virtual orbital. 

In my discussion of pyridine, I took a combination of these determinants that had the correct singlet spin symmetry (that is, the combination that represented a singlet state). I could equally well have concentrated on the triplet states. In modem Cl calculations, we simply use all the raw Slater determinants. Such single determinants by themselves are not necessarily spin eigenfunctions, but provided we include them all we will get correct spin eigenfunctions on diago-nalization of the Hamiltonian matrix. 

The relative importance of tlie different excitations may qualitatively be understood by noting tliat the doubles provide electron correlation for electron pairs, Quadruply excited determinants are important as they primarily correspond to products of double excitations. The singly excited determinants allow inclusion of multi-reference charactei in the wave function, i.e. they allow the orbitals to relax . Although the HF orbitals are optimum for the single determinant wave function, that is no longer the case when man) determinants are included. The triply excited determinants are doubly excited relative tc the singles, and can then be viewed as providing correlation for the multi-reference part of the Cl wave function. 

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