Multiple Determinants

A configuration interaction wave function is a multiple-determinant wave function. This is constructed by starting with the HF wave function and making new determinants by promoting electrons from the occupied to unoccupied or-... 

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

Electron correlation is often very important as well. The presence of multiple bonding interactions, such as pi back bonding, makes coordination compounds more sensitive to correlation than organic compounds. In some cases, the HF wave function does not provide even a qualitatively correct description of the compound. If the weight of the reference determinant in a single-reference CISD calculation is less than about 0.9, then the HF wave function is not qualitatively correct. In such cases, multiple-determinant, MSCSF, CASPT2, or MRCI calculations tend to be the most efficient methods. The alternative is... 

Johnson CD, Russell RL. A rapid, simple radiometric assay for cholinesterase suitable for multiple determinations. Anal. Biochem. 64 229-238, 1975. 

Wahlberg, J., and Spiess, M. (1997). Multiple determinants direct the orientation of signal-anchor proteins the topogenic role of the hydrophobic signal domain./ Cell... 

Note As each pair of signals delivers an independent mass value for the hypothetical singly charged ion, mass accuracy can greatly be enhanced in ESI by multiple determination of this value and subsequent calculation of the average. 

A point which may need emphasis, stated clearly in Hunter ( 2 ), is the precise interpretation of the confidence band about the predicted amount. This is important since without a clearly understood meaning, the interval will not be useful for assessing the precision of the predicted amounts or concentrations nor for comparing the results from various laboratories. Another reason the user of these methods must understand the interpretation is because increased precision can be achieved in at least two ways -by additional replication of the standards, which reduces the width of the confidence band about the regression line, and by performing multiple determinations on the unknowns, which reduces the width of the interval about the mean instrument response of the unknown. The interval for U is then given by... 

The coefficient of multiple determination, / is a measure of how much of SS, is accounted for by the factor effects. 

The coefficient of multiple determination ranges from 0 (indicating that the factors, as they appear in the model, have no effect on the response) to 1 (indicating that the factors, as they appear in the model, explain the data perfectly ). The square root of the coefficient of multiple determination is the coefficient of multiple correlation, R. 

Figure 9.4 Relationships among SS, SS and SS for calculating both the coefficient of multiple determination, R, and the variance-ratio for the significance of the factor effects,...

Calculate the coefficient of determination, /-j. and the coefficient of cdrrelation, r, for the model and data of Problem 9.6. What is the difference between the coefficient of determination and the coefficient of multiple determination ... 

Li, A.P. (2002) A review of the common properties of drugs with idiosyncratic hepatotoxicity and the multiple determinant hypothesis for the manifestation of idiosyncratic drug toxicity. Ctemico-Bioiogicoi Interactions, 142 (1-2), 7-23. 

Fig. 1.1. Examples for standard curves resulting from multiple determinations of different amounts of BSA. Line with circies protocol according to Lowry et al. Soiid iine. nonlinear regression dotted iine linear regressions wavelength 720 nm. Line with squares BCA protein determination. Soiid iine nonlinear regression dotted iine linear regression wavelength 562 nm). Findings means an example for graphical evaluation...

Carbenes with a triplet ground state well separated from the singlet (A st > 5 kcal/mol) react in a multiplicity determined by the method of generation. 

The chemical and physical requirements of the fuel shall conform to those listed in Table I, when tested in accordance with the applicable tests. Requirements contained herein are absolute and are not subject to correction for tolerance of test methods. However, if multiple determinations are made, average results shall be used... 

Depending on the R group, enthalpies of formation (AfT/m) shown in Table 2 have been obtained using either reaction and/or combustion calorimetry. Although there are fewer values for enthalpies of formation than for enthalpies of vaporization, there is still some confusion about the source, as well as the numerical value, of some of the data. To the extent possible, the sources have been obtained and the enthalpies of formation recorded in Table 2. Only the hquid phase enthalpies of formation are listed for those compounds that have multiple determinations of enthalpies of formation and of vaporization. 

So, the question arises of how we might modify the HF wave function to obtain a lower electronic energy when we operate on that modified wave function with the Hamiltonian. By the variational principle, such a construction would be a more accurate wave function. We cannot do better than the HF wave function with a single determinant, so one obvious choice is to construct a wave function as a linear combination of multiple determinants, i.e.. 

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