Error correlation

Several authors have been more modest in their goals and attempted to calculate directly thermodynamic quantities for reactions involving closed-shell species, where there is some hope that the correlation errors will cancel. The two papers often quoted in this field are those of Snyder and Basch (1969) and Hehre et al. (1970). 

Before discussing the correlation error, we will make some introductory remarks about the Hartree-Fock approximation based on the use of the Slater determinant (Eq. 11.38). We note that, if we... 

In Section II.C we gave a general discussion of the Coulomb correlation, and we will now define the correlation error in the independent-particle model in greater detail. It is convenient to study the first- and second-order density matrices and, according to the definitions (Eq. II.9) applied to the symmetryless case, we obtain... 

From Eq. 11.62 follows that the two-electron probability density does not show any "Coulomb hole" for pairs of either parallel or antiparallel spins, and this implies that the Hartree scheme is certainly affected by a large correlation error. 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

Our little table of accuracy shows that, in the Hartree scheme, there is a certain balance between the correlation errors for the two spin types which, in some cases, may lead to a cancellation of these errors with good theoretical results as a consequence. After the introduction of exchange, this balance is gone and, if the correlation between electrons with opposite spins is not taken into proper account, the final results may be influenced by this large... 

The ionization potential (7.9 eV) falls right outside the bracket of experimental IP s reported for carbon clusters with 40 to 100 atoms (6.42 eV IP 7.87 eV, Ref. 11). Inclusion of correlation effects will lower the calculated ASCF IP by 0.25 to 0.50 eV, so that the corrected IP will be at the upper end of the experimental IP>bracket. Due to the diffuseness of the n orbital from which an electron is removed, the correlation error in the ASCF value will be smaller than in cases where an electron is removed from a well localized bond. In these cases a correction of 1 eV is usually applied. 

It is common knowledge that, in the case of the hydrogen molecule studied in a minimal basis set, the correlation error can be explained by the existence of ionic species in the hydrogen dissociation products ... 

Cununing GL, Rollett JS, Rosotti FJC, Whewell RJ (1972) Statistical methods for the computation of stability constants, I. Straight-line fitting of points with correlated errors. J Chem Soc Dalton Trans 23 2652-2658... 

York D (1969) Least-squares fitting of a straight line with correlated errors. Earth Planet Sci Lett 5 320-324... 

Even an exact Hartree-Fock calculation cannot be exempt from the correlation error. A practical method of evaluation has been proposed by Hollister and Sinanoglu 16>. An LCAO SCF method has been applied to the calculation of the heat of various simple reactions by Snyder and Basch 17>. They have evaluated the correlation error by the method of Hollister and Sinanoglu 16>. 

Such a wave function is represented by a linear combination of wave functions for more than one electron configuration, and is called a "multi-configurational" wave function. The consideration of more than one configuration can reduce the correlation error. Such an approach is referred to as the method of configuration interaction (Cl) . 

Therefore, we consider the lOO CAS to cover most of the correlation effects and estimate the remaining correlation error to be not larger than about 1.5%. 

D.K. Melgaard and D.M. Haaland, Comparisons of prediction abilities of augmented classical least squares and partial least squares with realistic simulated data effects of uncorrelated and correlated errors with nonlinearities, Appl. Spectrosc., 58, 1065-73 (2004). 

Data point Feed K LV% Feed API Feed S Actual gasoline Gasoline predicted (Gary) Error % Gasoline predicted (Jones) Error % Gasoline predicted (Maple) Error % Gasoline predicted (correlation) Error % Gasoline predicted (NN) % Error... 

If we ignore the correlated errors, then from (8.32), we have... 

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