Hartree wavefunction

This expression is just the one which obtains for the Hartree product wave-function. The difference between this Hartree wavefunction and the Fock wavefunction of Eq. (1) is the absence of the antisymmetrizer j4 in that equation. This means that in the Hartree wavefunction each electron can be identified with a specific molecular orbital, whereas in the Fock wavefunction all electrons make use of all orbitals. The Hartree wavefunction is of course not a proper quantum mechanical wavefunction, since it is not antisymmetric in the electrons. Moreover, for the Fock wavefunction, it is in general not possible to reduce the interorbital exchange energy to zero. But the localized molecular orbitals, as defined here, represent that set of molecular orbitals for which the energy expression comes closest to the Hartree form, i.e. they come closest to being identifiable with electrons which are not exchanged among different orbitals. 

Due to the decoupled spin there is no difference between HF and Hartree wavefunctions in [GMU52] a different sign convention is applied to the 2p2 1Se function.) In these expressions one can see that the values of the mixing coefficients depend on the selected expansion. In addition, one can note relatively large contributions of components 0. Concentrating on the latter aspect, for simplicity it suffices to consider only the two electron configurations Is2 and 2p2, i.e.,... 

The Hartree wavefunction (above) is a product of one-electron functions called orbitals, or, more precisely, spatial orbitals these are functions of the usual space coordinates x, y, z. The Slater wavefunction is composed, not just of spatial orbitals, but of spin orbitals. A spin orbital ij/ (spin) is the product of a spatial orbital and a spin function, a or / The spin orbitals corresponding to a given spatial orbital are... 

The Hartree approximation is useful as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefunction is that it does not reflect the anti-S5mimetric nature of the electrons as required by the Pauli principle [7]. Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the summation in equation A1.3.11 does not include the /th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation Al. 3.11 solved for each orbital. 

Next, let us consider the Hartree wavefunction in Eq. (2.5). The symmetric and antisymmetric wavefunctions are given by... 

For atoms larger than He, the Hartree/SCF procedure is largely the same as for He. For an atom with N electrons, the Hartree wavefunction is... 

Using the single-particle orbitals of Eq. (2.57), we can write three possible Hartree wavefunctions ... 

In 1930, Fock pointed out that the Hartree wavefunction violates the Pauli Exclusion Principle because it is not properly... 

Using the orbitals, ct)(r), from a solution of equation Al.3.11, the Hartree many-body wavefunction can be constructed and the total energy detemiined from equation Al.3,3. 

Field M J 1991 Constrained optimization of ab initio and semiempirical Hartree-Fock wavefunctions using... 

The total wavefunction r2,. . ., r is written as a product of single-particle functions (Hartree approximation). The various integrals are evaluated in tire saddle point approximation. A simple Gaussian fomr for tire trial one-particle wavefunction... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

Hartree-Fock wavefunction, is an eigenfunction of and the corresponding oth-order energy Eg° is equal to the sum or orbital energies for the occupied molecular... 

To obtain an improvement on the Hartree-Fock energy it is therefore necessary to use Moller-Plesset perturbation theory to at least second order. This level of theory is referred to as MP2 and involves the integral J dr. The higher-order wavefunction g is... 

Unfortunately, this only holds for the exact wavefunction and certain other types ol leavefuiiction (such as at the Hartree-Fock limit). Moreover, even though the Hellmarm-Feynman forces are much easier to calculate they are very unreliable, even for accurate wavefunctions, giving rise to spurious forces (often referred to as Pulay forces [Pulay l )S7]). 

The predicted energy, which appears in the SCF summary section preceding the stability analysis output, is -149.61266 hartrees, which is about 53.5 kcal/mol lower than that corresponding to the RHF wavefunction (-149.52735). 

In order to specify the proper electronic state, ozone calculations should be performed as unrestricted calculations, and the keyword Gue s=Mix should always be included. This keyword tells the program to mix the HOMO and LUMO within the wavefunction in an effort to destroy a-P and spatial symmetries, and it is often useful in producing a UHF wavefunction for a singlet system. Running a UHF GuesssMix Stable calculation confirms that the resulting wavefunction is stable, and it predicts the same energy (-224.34143 hartrees) as the previous Stable=Opt calculations. 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

However, such a function is not antisymmetric, since interchanging two of the r, s —equivalent to swapping the orbitals of two electrons—does not result in a sign change. Hence, this Hartree product is an inadequate wavefunction. 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

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. 

The full Cl method forms the wavefunction as a linear combination of the Hartree-Fock determinant and all possible substituted determinants ... 

There are m doubly occupied molecular orbitals, and the number of electrons is 2m because we have allocated an a and a spin electron to each. In the original Hartree model, the many-electron wavefunction was written as a straightforward product of one-electron orbitals i/p, i/ and so on... 

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