Molecular orbitals that are linear combinations

LCAOMOs. Approximate Molecular Orbitals That Are Linear Combinations of Atomic Orbitals... 

The exact Born-Oppenheimer orbitals for hJ are expressed in an unfamiliar coordinate system and we did not explicitly display them. It will be convenient to have some easily expressed approximate molecular orbitals that resemble the correct molecular orbitals. We define molecular orbitals that are linear combinations of atomic orbitals, abbreviated LCAOMOs. If /i, f2, /s,... are a set of basis functions, then a linear combination of these functions is written as in Eq. (16.3-34) ... 

The orbital has its orbital region centered at nucleus A and the orbital has its orbital region centered at nucleus B. We now form molecular orbitals that are linear combinations of these two basis functions ... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

A molecular orbital is a linear combination of basis functions. Normalization requires that the integral of a molecular orbital squared is equal to 1. The square of a molecular orbital gives many terms, some of which are the square of a basis function and others are products of basis functions, which yield the overlap when integrated. Thus, the orbital integral is actually a sum of integrals over one or two center basis functions. 

Molecular Orbital Theory Model. Oxygen and hydrogen atoms in H2O are held together by a covalent bond. According to the quantum molecular orbital theory of covalent bonding between atoms, electrons in molecules occupy molecular orbitals that are described, using quantum mechanical language, by a linear combination of... 

The basic concept of molecular orbital theory is that molecular orbitals may be constructed from a set of contributing atomic orbitals such that the molecular wave functions consist of linear combinations of atomic orbitals (LCAO).. in the case of the combination of two Is hydrogen atomic orbitals to give two molecular orbitals, the two linear combinations are written below so that atomic wave functions are represented by j/ and molecular wave functions by <)> ... 

Obviously, an orbital boundary surface defines an interior and an exterior. Outside the boundary, the function cp has very small values because its square, summed over all space from the boundary wall to infinity, has a value of only 0.1. Recognizing this fact allows the LCAO approximation to be interpreted in physical terms. When we say that a molecular orbital is a linear combination of AOs, we imply that it is almost indistinguishable from cpk in the neighbourhood of atom k. This is because we are then inside the boundary of cpk and outside the boundary oi(pfl = k), so that cpk has finite values and contributions from [Pg.24]

The theory (7, 8, 9,10,11,12) will be outlined for molecules having n atoms with a total of P valence shell electrons. We seek a set of molecular orbitals (LCAO-MO s), that are linear combinations of atomic orbitals centered on the atoms in the molecule. Since we shall not ignore overlap, the geometry of the molecule must be known, or one must guess it. The molecule is placed in an arbitrary Cartesian coordinate system, and the coordinates of each atom are determined. Orbitals of the s and p Slater-type (STO) make up the basis orbitals, and as indicated above we restrict ourselves to the valence-shell electrons for each of the atoms in the molecule. The STOs have the following form for the radial part of the function (13,18) ... 

The second comment which it is appropriate to make here is that we can already see the reason for o-k separation by this method. In the LCAO-scheme we represent a molecular orbital as a linear combination of atomic orbitals, as per equation (2-1). We are going now to ask what kind of conclusion would be reached if, in the summation of equation (2-1), we supposed that some of the were u-orbitals, in addition to the orbitals of Ji-symmetry... 

The tight-binding quantum-chemical model with molecular orbitals that are a linear combination of atomic orbitals provides a reasonable description of the delocalization of the d valence electrons. The s,p valence electrons are well described as free electrons [11]. 

We first assume - like Lewis - that the electrons in inner shells are unaffected by bond formation, i.e. that two electrons continue to occupy the Li 1 orbital. Secondly we assume that the two valence electrons occupy bonding molecular orbitals formed by linear combination of the two AOs that contain the valence electrons when the two atoms are far apart, i.e. by combination of the Li 2s and the H l.y orbitals, which we denote by [Pg.115]

In the molecule of buta-1,3-diene, there are four p-orbitals located on four adjacent carbon atoms and hence this generates four new tt -molecular orbitals on overlapping. The way to get these new 7r-molecular orbitals is the linear combination of two ir-molecular orbitals of ethene according to the perturbation molecular orbital (PMO) theory. Like the combination of atomic orbitals, overlapping of the bonding (ct or tu) or antibonding molecular orbitals (a or TT ) of the reactants (here, ethene) results in the formation of the new molecular orbitals that are designated as Wi, W2, etc. in the product (here, buta-1,3-diene). 

A molecular orbital diagram for a transition metal complex can be generated from the orbitals of the metal and the symmetry-adapted linear combinations (SALCs) of the orbitals of the ligands. The SALCs are typically illustrated on one side of the diagram, the orbitals of the metal on the other side, and the molecular orbitals that result from combining the... 

Basis function-functions describing the atomic orbitals that when linearly combined make up the set of molecular orbitals in a quantum mechanics calculation Gaussian basis sets and Slater type orbitals are examples of basis functions. 

In contrast to VBT, "full-blown" MOT considers the electrons in molecules to occupy molecular orbitals that are formed by linear combinations (addition and subtraction) of all the atomic orbitals on all the atoms in the structure. In MOT, electrons are not confined to an individual atom plus the bonding region with another atom. Instead, electrons are contained in MOs that are highly delocalized—spread across the entire molecule. MOT does not create discrete and localized bonds between neighboring atoms. An immediate benefit of MOT over VBT is its treatment of conjugated tt systems. We don t need a "patch" like resonance to explain the structure of a carboxylate anion or of benzene it falls naturally out of the delocalized nature of the MOs. The MO models of simple molecules like ethylene or formaldehyde also lead to bonding concepts that are pervasive in organic chemistry. 

We must represent the molecular orbitals in our calculations in a manner such that they can be incrementally changed in the SCF process. A common approach is to represent a molecular orbital as a linear combination of atomic orbitals (Eq. 14.33). Here, the subscript / is the same as in the HE equation, where i is a, b, c, etc. The subscript k stands for all different atomic orbitals that are included on the atoms in the molecule. Hence, every MO can potentially have a contribution from every AO in the molecule. 

The bonds in polyatomic molecules are built in the same way as in diatomic molecules, the only differences being that we use more atomic orbitals to construct the molecular orbitals and these molecular orbitals spread over the entire molecule, not just the adjacent atoms of the bond. In general, a molecular orbital is a linear combination of all the atomic orbitals of all the atoms in the molecule. 

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