Density LCAO approximation

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also... 

Figure 8.IB shows an experimental contour map of electron density for the H2O molecule in plane y-z, after Bader and Jones (1963). The electron density is higher around the nuclei and along the bond directrix. The experimental electron density map conforms quite well to the hybrid orbital model of Duncan and Pople (1953) with the LCAO approximation.

The electronic charge density in an MO extends over the whole molecule, or at least over a volume containing two or more atoms, and therefore the MOs must form bases for the symmetry point group of the molecule. Useful deductions about bonding can often be made without doing any quantum chemical calculations at all by finding these symmetry-adapted MOs expressed as linear combinations of AOs (the LCAO approximation). So we seek the LCAO MOs... 

Let s consider the shape of the MO first. The simplest picture considers molecular orbitals as resulting from the overlap of atomic orbitals. When atoms are separated by their usual bonding distance, their AOs overlap. Where this overlap occurs, either the electron waves reinforce and the electron density increases, or the electron waves cancel and the electron density decreases. The left-hand side of Figure 3.3 shows the overlap of the Is atomic orbitals on two different hydrogens (Hu and H/ ) when these hydrogens are separated by their normal bonding distance. The two atomic orbitals interact to produce two molecular orbitals. The MOs result from a linear combination of the AOs (called the LCAO approximation). Simply, this means that the AOs are either added (lsa + lsfc) or subtracted (1 sa — lst) to get the MOs. 

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

In order to perform a qualitative analysis of the /1-decay-induced redistribution of electron density it is sufficient to calculate the molecular electron states in the MO LCAO approximation, i.e., not taking into account the correlation of electrons. Below we present the calculation data for a number of molecules, which we have obtained using the Gaussian-70 program with the basis of s and p functions. We have used the extended atomic basis 4-31G, which contains about twice as many atomic functions as the minimal one (Ditchfield et al, 1971). 

The electron density distribution was calculated according to Mulliken (1955 see also Herzberg, 1966). In the MO LCAO approximation the ith molecular orbital is... 

The quantum mechanical indices described above are obtained from calculated molecular wavefunctions. The quality of the wave-function and, consequently, of the indices depends entirely on the formalism and the level of approximation one uses. Because the molecular wavefunction is often described as a linear combination of atomic orbitals (LCAO), one can easily obtain some of the indices mentioned above. These include the atomic net charges, sigma and i charges, frontier electron densities, Euqmq and LEMO t ie superdelocalizability parameters. Within the LCAO approximation one can assign the reactivity indices to specific atoms or bonds in the molecule. These indices reflect the stationary reactivity properties of the atoms and bonds in the molecule as described by charges or orbital energies and can therefore serve only as indicators of the reactivity. 

In the forms (l)-(4) the density matrices are expressed in terms of the continuous variables which define the positions of the electrons. But if, as often happens, our wavefunction is compounded out of atomic orbitals (e.g., the LCAO approximation), we may find it instructive to use an AO or an MO base. An example that we shall need later is the MO ground state of butadiene, where, in the 7r-electron approximation, we fill the two mutually... 

The Linear Combination of Atomic Orbitals (LCAO) approximation is fundamental to many of our current models of chemistry. Both the vast majority of the calculational programs that we use, be they ab initioy density functional, semiempirical molecular orbital, or even some sophisticated force-fields, and our qualitative understanding of chemistry are based on the concept that the orbitals of a given molecule can be built from the orbitals of the constituent atoms. We feel comfortable with the Ji-HOMO (Highest Occupied Molecular Orbital) of ethylene depicted as a combination of two carbon p-orbitals, as shown in Fig. 2.1, although this is not a very accurate description of the electron density of this Molecular Orbital (MO). The use of the Jt-Atomic Orbitals (AOs), however, makes it easier to understand both the characteristics of the MO itself and the transformations that it can undergo during reactions. 

In principle, we could build up MOs from many sorts of function that can describe an electron density probability distribution, but we have learnt to understand combinations of AOs and to use them in our models of chemical bonding and reactivity. Indeed, if confronted with an MO that was calculated, for instance, using plane waves, most chemists would immediately translate it into a combination of AOs. In the following, we will describe the LCAO approximation and demonstrate some of the effects that are important when AOs interact with each other to form MOs. 

The simplest example of the LCAO approximation is the combination of two s-AOs to form s- (bonding) and (antibonding) MOs. This is shown in Fig. 2.2 for the dihydrogen molecule. Figure 2.2a shows the simple orbital interaction diagram whereas Fig. 2.2b shows 3D-electron density contour plots for the AOs and MOs. 

The spinless one-electron density matrix (DM) elements are defined in the LCAO approximation as... 

The electron energy of the crystal (per primitive unit cell) as calculated within the HF LCAO approximation can be expressed in terms of the one-electron density matrix (DM) and includes the lattice sums... 

The One-electron Density Matrix of the Crystal in the LCAO Approximation... 

The electronic structure for the MgO crystal was calculated in [608] both in the LCAO approximation and in the PW basis. In both cases the calculations were done by the density-functional theory (DFT) method in the local density approximation (LD A). The Monkhorst Pack set of special points of BZ, which allows a convergence to be obtained (relating to extended special-points sets) in the calculations of electronic structure, was used in both cases. For the LCAO calculations the Durand Barthelat pseudopotential [484] was used. In the case of the PW calculations the normconserving pseudopotential and a PW kinetic energy cutoff of 300 eV were used. 

The perturbed-cluster approach [701,702] is formulated in the LCAO approximation and is based on the following sequence of steps, [703] a) subdivide the entire defect system into a molecular cluster (C), containing the defect, and an external region (D), the indented crystal b) calculate the wavefunction for the molecular cluster in the field of the indented crystal c) correct the cluster solution in order to allow for the propagation of the wavefunction into the indented crystal while generating the density matrix of the defect system. Steps b and c are repeated to self-consistency. The corrective terms in step c are evaluated by assuming that the density of states projected onto the indented crystal is the same as in the perfect host crystal (fundamental approximation). 

In the linear combination of atomic orbitals (LCAO) approximation of the molecular orbital, the energy of the overlap electron density between the atomic orbitals Xu and Xv due to attraction by the core. See Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Relativistic Effective Core Potential Techniques for Molecules Containing Very Heavy Atoms Transition Metal Chemistry and Transition Metals Applications. 

Fill in the details of the derivation of the spin-spin coupling formula (11.7.13). Obtain the coupling function Qss(dd ri, rj) for the standard state a with M = S, using a 1-determinant wavefunction with the spins of the open-shell electrons all parallel-coupled and show that this function is then expressible in terms of the spin-density matrix Qs da r, r0- What kind of integrals will you have to evaluate to obtain, in an LCAO approximation, the tensor components that determine the zero-field splitting of Zeeman levels [Hint Closed-shell electrons give no contribution to Qs or Qss, while for electrons in orbitals with a common a factor the 2-electron density matrix tr is related to p as in Section 5.3. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

The fundamental tool for the generation of an approximately transferable fuzzy electron density fragment is the additive fragment density matrix, denoted by Pf for an AFDF of serial index k. Within the framework of the usual SCF LCAO ab initio Hartree-Fock-Roothaan-Hall approach, this matrix P can be derived from a complete molecular density matrix P as follows. 

---