Wavefunctions described

Eyi. This is obviously not the case, and analysis of the results shows that the HF wavefunction describes the following process, for large R ... 

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

One consequence of this annihilation algorithm is that the number of atoms involved in a specific LMO increases as a result of mixing the original LMOs. The wavefunction describing the new occupied LMOs not only has intensity on atoms originally in the LMO but also on atoms with which the virtual LMO was associated. If no action is taken, then the number of atoms spanned by a given LMO increases until every LMO includes contributions, albeit extremely small ones in most instances, from every atom in the QM system. As a consequence, after each iteration to solve the SCF equations, the contributions to each LMO from individual atoms are examined, so that if those associated with a specific atom, J, are small, then atom J is deleted from the LMO. In practice, the number of atoms that contribute to LMOs appears to reach a limit of 100-130, as the number of atoms in the molecule increases. 

The phasing of the molecular orbitals (shown as +/-) is a result of the wavefunctions describing the orbitals. + shows that the wavefunction is positive in a particular region in space, and - shows that the wavefunction is negative. 

The Xi and %2 wavefunctions describe the motion of particles of total energy W/4 in the potentials F( )/4 or V rj)l4. It is also interesting to note that for the case E=0 the wave equations of Eqs. (6.22a) and (6.22b) are similar to the radial equation for the coulomb potential in spherical coordinates. Explicitly, making the substitutions... 

In the atomic system of units, the energy of a ground-state hydrogen atom is —j Eh and so we would expect the potential curve to tend asymptotically to -ljEh This is obviously not the case, and analysis of the results shows that the HF wavefunction describes the following process, for large R ... 

Here (4 f) is/the total wavefunction describing the initial (final) state of the system and is an eigenfunction the... 

The "crude" adiabatic method enables us to develop another approach. If the translational motion is slower than the internal motion (a << 1) then the D state is described by eqs. 7A, 76, and 81-83. The wavefunction describing the opposite case (a ... 

As observed in many theoretical studies of symmetric radicals (92), a HF wavefunction describing it ->- tt excited states preferentially localizes the hole on one moiety leading to a broken symmetry structure of the states with unequal bond lengths for the minimum energy structure. Similarly, for the n tt ... 

Furthermore, the wavefunction describing the bond direction for quantum numbers j and ni is identical to the angular portion of the hydrogen orbitals for quantum numbers / and in/. So the ground state j = m j = 0 is spherically symmetric, just like an s orbital the j = 1 states look like p orbitals, and so on. 

The wavefunction plays a central role in quantum mechanics. For atomic systems, the wavefunction describing the electronic distribution is called an atomic orbital in other words, the aforementioned Is wavefunction of the ground state of a hydrogen atom is also called the Is orbital. For molecular systems, the corresponding wavefunctions are likewise called molecular orbitals. 

Thus, the simplest wavefunction describing say three bonds using the Valence Bond model in addition to an orthogonal core is given by... 

In the MO formalism it is quite straightforward to deal with the excited states of a molecule. An adequate wavefunction of an excited state can be constructed according to the resultant configuration and its symmetry arising from electron promotion among MO series. Compared with numerous MO-based methods, VB approaches are far less employed to study excited states due to the difficulty in VB computations. Recently, by observing the correlation between MO theory and resonance theory, as well as the symmetry-adapted VB wavefunction described in the last section, we performed VB calculations on low-lying states of some molecules [71, 72],... 

What is obviously missing from the seven-configuration wavefunction described in the previous Subsection is some correlation across the plane of the nuclei for the two inner orbitals, analogous to that already included for the three valence pairs. The most economical way to provide this missing correlation is of course through the addition of a fully-symmetric configuration consisting of three symmetry-equivalent pairs of a orbitals and a pair of identical C3- and <7v-invariant n orbitals. 

AFM ground state. An unasked question is At what x does the chosen RVB variational wavefunction have a lower energy than a wavefunction describing AFM at appropriate wave-vectors Q(jc) They also acknowledge that their approach is no help in understanding the universal normal state properties for compositions near those for the highest Tc. I will return to both these points first, let us compare the claims made with the experiments. 

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