Orbitals of hydrogen atom

Figure 2.1 Radial probability density plots for Is and 2s orbitals of hydrogen atom...

The combination of 1 s orbitals of hydrogen atoms to form molecular orbitals has been shown in figure below. 

In Fig. 11.4, the log-and 1ct -orbitals are plotted as functions of z, along the internuclear axis. Both functions have cusps, discontinuities in slope, at the positions of the two nuclei A and B. The I5 orbitals of hydrogen atoms have these same cusps. The shape of the Icrg and 1[Pg.86]

The AO of the two boron atoms, f (2 ) and f (B2) which are approximately sp3 and the orbital of hydrogen atom f (H) which is an s orbital have similar energy and the appreciable spatial overlap, but it is only the combination f (B ) + f (B2) which has the correct symmetry to combine linearly with f (.H). The three normalized and orthogonal MOs have the approximate form. 

Figure 12-4 shows the formation of C2H4 from two 5p -hybridized carbon atoms and four hydrogen atoms. Two sp orbitals of each carbon form tr-bonds with I5 orbitals of hydrogen atoms. Overlap between the remaining sp orbital of each of the two carbons forms the stronger double bond. The weaker ir-bond of the double bond is formed by overlap of the 2p orbitals of the two carbons. This overlap is indicated by the dashed lines. The rr-bond is relatively weak since the 2 p orbitals are perpendicular to the plane of sp orbitals and can only overlap in a sidewise manner. [Strong overlap of 2p orbitals requires frontal (colinear) overlap as in the formation of the bond in F2 —see Problem 3.11.]... 

The function /(r) (in atomic units) for 2p orbitals of hydrogenic atoms is... 

Of course, the answer is no. The absolute limit M is equal to half the number of the eleetrons. beeause only then ean we create N spinorbitals and write the Slater determinant. However, in quantum chemistry (rather misleadingly), we call the minimum basis set the basis set resulting from an inner shell and valence oibitals in the corresponding atoms. For example, the minimum basis set for a water molecule is I5,2s and three 2p orbitals of oxygen and two I5 orbitals of hydrogen atoms, which is seven AOs in total (while the truly minimal basis would contain only 10/2 = 5 AOs). 

Fig. 23 Molecular orbital of the hydrogen molecule, , composed of the atomic orbitals of hydrogen atoms,...

The Hiickel MO method assumes that the a and n systems can be treated independently and then considers only the n system. The extended Hiickel theory (EHT), developed by Hoffmann, considers all of the valence electrons in the system, but it does not deal with the core electrons (those in orbitals below the valence shell). ° The EHT wave function is then written as a linear combination of functions describing each type of valence orbital on each atom in the structure. For hydrocarbons, the EHT wave fimctions include the 2s and all three 2p atomic orbitals of carbon as well as the Is orbitals of hydrogen atoms. The overlap integrals (S,y) are computed, whereas they were ignored in HMO theory. The elements of the secular determinant are assigned a value... 

It has been proved that each of the Hartree-Fock orbitals has the same asymptotic dependence on the distance from the molecule (N.C. Handy, M.T. Matron, HJ. Silverstone,PM x. Rev. 180 (1969) 45), i.e. const - exp( —V—2Ema. r), where Cma. is the orbital energy of HOMO. Earlier, people thought the orbitals decay as exp(—2c,-r), where c,- is the orbital energy expressed in atomic units. The last formula, as is easy to prove, holds for the atomic orbitals of hydrogen atoms (see p. 178). R. Ahlrichs,... 

Scalar relativistic effects increase steadily when one goes down the periodic table. They are essentially negligible for HF but become moderately important for HBr and more so for HI. Since the H-X bond is composed of a H orbital of hydrogen atom and s-p hybridized orbitals of halogen atom, scalar relativistic effects generally induce a bond contraction due to the relativistic spatial contraction of i-type and p-type orbitals. The relativistic shortening of the bond distance becomes manifest for HBr and amounts to around 0.006 Bohr. For HI, it results in a 0.02 Bohr shortening of bond. The scalar relativistic effects are relatively small for other parameters (e.g., they amount to a few parts per million for harmonic frequencies). 

The C—H bonds in methane, CH4, are described by valence bond theory as the overlapping of each sp hybrid orbital of the carbon atom with Is orbitals of hydrogen atoms (see Figure 10.22Q. Thus, the bonds are arranged tetrahedrally, which is... 

Hybrid Orbitals Bonds formed by the overlap of the 1 s orbitals of hydrogen atoms and the sp orbitals of (a) carbon, (b) nitrogen, and... 

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