Numbers of Electrons and Orbitals

This argument of the spatial overlap of orbitals can be extended to any number of electrons and orbitals and is the basis of Hund s rule. 

The critical choice was made of a CASSCF(4,4)/6-31G calculation the active space is thus the degenerate filled 2s + 2s and 2s 2s pair of MOs, and the degenerate empty 2px + 2px and 2px — 2px pair of MOs. CASSCF(4,4) was chosen because it corresponds to the CASSCF(2,2) calculation on one beryllium atom in the sense that we are doubling up the number of electrons and orbitals in our noninteracting system. This calculation gave an energy of 29.1709451 hartree. We can compare this with twice the energy of one beryllium atom, 2 x — 14.5854725 hartree = —29.1709450 hartree. 

Despite these restrictions, the GVB and SC methods generally provide energies that are much closer in quality to CASSCF than to Hartree Fock (19), and wave functions that are close to the CASSCF wave function having the same number of electrons and orbitals in the active space. This property has been used to devise a fast method to get approximate SC wave functions. 

It is obvious that ECPs address the size problem directly, by reducing the number of electrons and orbitals in the calculations. Effective core potentials can also be used, albeit somewhat indirectly, to address the electron correlation challenge. By reducing resources needed for other parts of the computational exercise, one makes it possible to increase the focus on electron correlation. When it is necessary to include electron correlation, the computational effort can be proportional to potentially further limiting the size (and thus to... 

As it will be seen in next Section, metal cluster structures are closely related to the number of electrons and orbitals existing in the cluster core. In general. 

The basis of the Skeleton Electron Pair Theory described above rest on the assumption that the constituent fragments of main group and transition metal clusters contribute both the same number of electrons and orbitals for the cluster bonding and that these orbitals have similar nodal properties. 

If the number of orbitals increases, the first quantized derivation presented above becomes more and more involved. As the size of the determinant increases, one should apply the general Slater-Condon rules, or rederive them similarly as was done for the Hiickel energy expression (Sect. 6.4). On the other hand, the general result of Eq. (6.23) is valid for any number of electrons and orbitals. 

In the time-consuming step of the A/ -resoIution method, we did not exploit the separation of excitation operators into alpha and beta spin parts as in (11.7.10). We shall now consider a different method that more fully exploits the separation into the alpha and beta spin spaces. This algorithm is known as the minimal operation-count (MOC) method [4] since it yields an operation count that, to leading orders in the numbers of electrons and orbitals, is identical to the theoretical minimum (11.7.16). 

Since both molecules have the same number of electrons, the orbital numbered 8 is the HOMO, and the one numbered 9 is the LUMO in both cases. However, they are not the same type orbitals. Let s consider ethylene first. 

A CASSCF calculation is requested in Gaussian with the CASSCF keyword, which requires two integer arguments the number of electrons and the number of orbitals in the active space. The active space is defined assuming that the electrons come from as many of the highest occupied molecular orbitals as are needed to obtain the specified number of electrons any remaining required orbitals are taken from the lowest virtual orbitals. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

A canonical set of structures for a system with more orbits than electrons is obtained by arranging all the orbits (including phantom orbits for 5>0) in a ring and then drawing non-intersecting bonds to a number determined by the number of electrons and the multiplicity. If two electrons occupy the same orbit, forming an unshared pair, a loop is drawn with its ends at the orbit. 

Using the fact that the energy is linear with respect to the number of electrons and Janak s theorem [31], the orbital energies of the N—n and N+n electron system become equal to the exact ground state vertical ionization energy and electron affinity, respectively ... 

Most stable ground-state molecules contain closed-shell electron configurations with a completely filled valence shell in which all molecular orbitals are doubly occupied or empty. Radicals, on the other hand, have an odd number of electrons and are therefore paramagnetic species. Electron paramagnetic resonance (EPR), sometimes called electron spin resonance (ESR), is a spectroscopic technique used to study species with one or more unpaired electrons, such as those found in free radicals, triplets (in the solid phase) and some inorganic complexes of transition-metal ions. 

The pyrylium cation is isoelectronic with pyridine it has the same number of electrons and, therefore, we also have aromaticity. Oxygen is normally divalent and carries two lone pairs. If we insert oxygen into the benzene ring structure, then it follows that, by having one electron in a p orbital contributing to the aromatic sextet, there is a lone pair in an sp orbital,... 

Using the above definitions for the four quantum numbers, we can list what combinations of quantum numbers are possible. A basic rule when working with quantum numbers is that no two electrons in the same atom can have an identical set of quantum numbers. This rule is known as the Pauli Exclusion Principle named after Wolfgang Pauli (1900-1958). For example, when n = 1,1 and mj can be only 0 and m can be + / or -1/ This means the K shell can hold a maximum of two electrons. The two electrons would have quantum numbers of 1,0,0, + / and 1,0,0,- /, respectively. We see that the opposite spin of the two electrons in the K orbital means the electrons do not violate the Pauli Exclusion Principle. Possible values for quantum numbers and the maximum number of electrons each orbital can hold are given in Table 4.3 and shown in Figure 4.7. 

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