Wavefunction exact

To prove this we expand i about the exact wavefunction i that is, we let... 

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]). 

In other words, the energy of the exact wavefunction serves as a lower bound to the energies calculated by any other normalized antisymmetric function. Thus, the problem becomes one of finding the set of coefficients that minimize the energy of the resultant wavefunction. 

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. 

J(d i>/da)H l> dr were therefore zero, as substitution of P = e P into nation 14.18 might suggest. Unfortunately, exact wavefunctions are very hard tg come by and only for an exact wavefunction. 

Such a calculation with exact wavefunctions shows ... 

The electron- and spin-densities are the only building blocks of a much more powerful theory the theory of reduced density matrices. Such one-particle, two-particle,. .. electron- and spin-density matrices can be defined for any type of wavefunction, no matter whether it is of the HF type, another approximation, or even the exact wavefunction. A detailed description here would be inappropriate... 

A different way to approximate SS-MRCC is the so-called externally corrected CCSD (ec-CCSD) [13-17], The ec-CCSD method is based on the Coupled Cluster Approach (CCA). In CCA the exact wavefunction is written in an exponential form,... 

Rosina s theorem states that for an unspecified Hamiltonian with no more than two-particle interactions the ground-state 2-RDM alone has sufficient information to build the higher ROMs and the exact wavefunction [20, 51]. Cumulants allow us to divide the reconstruction functional into two parts (i) an unconnected part that may be written as antisymmetrized products of the lower RDMs, and (ii) a connected part that cannot be expressed as products of the lower RDMs. As shown in the previous section, cumulant theory alone generates all of the unconnected terms in RDM reconstruction, but cumulants do not directly indicate how to compute the connected portions of the 3- and 4-RDMs from the 2-RDM. In this section we discuss a systematic approximation of the connected (or cumulant) 3-RDM [24, 26]. 

The Hamiltonian gradually filters the ground-state wavefunction from the trial wavefunction. To understand this filtering process, we expand the initial trial wavefunction in the exact wavefunctions of the Hamiltonian, ). With n iterations of the power method, we have... 

In the method based on the unitary transformation, we start by writing the exact wavefunction th in terms of the reference function and a unitary transformation operator in Fock space ... 

Compared to the original particles, the new particles associated with c, cj have been dressed by the dynamic correlations in U. Consequently, we can represent the exact wavefunction as a simpler function of the new particles. 

As Eqs. (18) and (2) demonstrate, the exact wavefunction is no longer explicitly needed nor is an approximate wavefunction used. Instead, the generating geminals determine the wavefunction as well as the reduced density matrices and thereby the energy of an AGP wavefunction. 

In quantum chemistry, the correlation energy Ecorr is defined as Econ = exact HF- In Order to Calculate the correlation energy of our system, we show how to calculate the ground state using the Hartree-Fock approximation. The main idea is to expand the exact wavefunction in the form of a configuration interaction picture. The first term of this expansion corresponds to the Hartree-Fock wavefunction. As a first step we calculate the spin-traced one-particle density matrix [5] (IPDM) y ... 

Figure 4. Two-site Hubbard model. Upper curve is the entanglement calculated by the von Newmann entropy. The curves 5 1 and 5 2 are the correlation entropies of the exact wavefunction as defined in the text. The dashed line is the 5 2 for the combined wavefunction based on the range of V values. S for the combined wavefunction is zero.

Here, V is a small perturbation and X, is a dimensionless parameter. Expanding the exact wavefunction and energy in terms of the Hartree-Fock wavefunction and energy yields. 

Another approach to providing atomic charges is to fit the value of some property which has been calculated based on the exact wavefunction with that obtained from representation of the electronic charge distribution in terms of a collection of atom-centered charges. In practice, the property that has received the most attention is the electrostatic potential, 8p. This represents the energy of interaction of a unit positive charge at some point in space, p, with the nuclei and the electrons of a molecule (see Chapter 4). 

Fig. 7.4. Wavefunctions of the hydrogen molecular ion. (a) The exact wavefunctions of the hydrogen molecular ion. The two lowest states are shown. The two exact solutions can be considered as symmetric and antisymmetric linear combinations of the solutions of the left-hand-side and right-hand-side problems, (b) and (c), defined by potential curves in Fig. 7.3. For brevity, the normalization constant is omitted. (Reproduced from Chen, 1991c, with permission.)...

The exact wavefunction corresponds to a = b = s and a = (3 = Z. The variational ground state energy of Z = 90 hydrogen-like ion in the Dirac-Pauli... 

When the variational method is applied to the functional (3.1) the convergence of the energy E is particularly efficient and much faster than the convergence of other properties which can be derived from the same wavefunction. This can be seen by the following set of operations. Say that the approximate wavefunction ip has a small error A which can be chosen orthogonal to the exact wavefunction. The energy functional (3.1) can then be written,... 

For the GS, the HK theorems42 guarantee thatEq. (10) of different exact theories all deliver the same GS density in spite of distinct mathematical structures of Oeft (r [p]) within different theoretical approaches58-60 (i.e. local vs. nonlocal operators). The reason is simple the density is one-to-one mapped on to the GS wavefunction, regardless of how the exact wavefunction and the exact density are calculated. 

We conclude that KS orbitals seem to be just as suitable, if not better, for qualitative MO theoretical considerations than other orbitals, e.g., HF orbitals. The KS orbitals offer the advantage, in particular over semiempirical orbitals, but also over HF, that they are connected in an interesting way with the exact wavefunction and with exact energetics. So the MO-theoretical analysis put forward in the next section deals with energetic contributions that sum up to the exact or, with the present state of the art in density functionals, at least accurate interaction energy. The KS model offers an MO-theoretical universe of discourse in which molecular energetics can be interpreted in terms of considerations that until now were necessarily inaccurate and qualitative. Is this MO-... 

The other approach most frequently used to describe a correlated wavefunction beyond the independent-particle model is based on configuration interaction (Cl). (If the expansion is made on grounds of other basis sets, the approach is often called superposition of configurations, SOC, in order to distinguish it from the Cl method.) According to the general principles of quantum mechanics, the exact wavefunction which is a solution of the full Hamiltonian H can be obtained as an expansion in any complete set of basis functions which have the same symmetry properties ... 

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