Orbitals in hydrogen

FIGURE 1.32 The radial distribution function tells us the probability density for finding an electron at a given radius summed over all directions. The graph shows the radial distribution function for the 1s-, 2s-, and 3s-orbitals in hydrogen. Note how the most probable radius icorresponding to the greatest maximum) increases as n increases. 

Electronic and nuclear energy in H2. a. Values for non-interacling electrons. 6, Coulomb energy of nuclear repulsion, c, Approximate electronic energy curve for interacting electrons. Units ordinates, 1 = Rydberg constant, abscissas, 1 = radius of first Bohr orbit in hydrogen atom. 

Although mixing of s and p orbitals is represented in Fig. 5.24 as a sepamle step preceding the formation of molecular orbitals, the entire process can be combined into a single step. For example, the bonding molecular orbital in hydrogen chloride may be considered to be formed as... 

To turn again momentarily from the abstractions of orbitals and quantum numbers back to the spectra that generated them, consider the transition from a Is orbital In hydrogen to a 2p orbital. The terms and transitions are ... 

FIGURE 10.2 Molecular orbitals in hydrogen fluoride, plotted using two isosurfaces, at 0.02 e 1/2 Bohr 3/2. (See the color version of this figure in Color Plates section.)... 

Relative energies of atomic and molecular orbitals in hydrogen molecule are shown in figure below. 

Which is higher in energy the 2s or 2p orbital in hydrogen Is this also true for helium Explain. 

Fig. 39. Atomic orbitals in hydrogen fluoride. The fluorine 2p and 2p atomic orbitals, normal to the axis, are omitted for clarity.

We must now look at the coefficients, c, of equation 2-1. When there are electrons in the orbital, the squares of the c-values are a measure of the electron population in the neighbourhood of the atom in question. Thus in each orbital the sum of the squares of all the c-values must equal one, since only one electron in each spin state can be in the orbital. Now the orbitals in hydrogen are symmetric about the mid-point of the H—H bond in other words c2 must equal c2. Thus we have defined what the values of c, and c2 in the bonding orbital must be, namely 1/ /2 = 0-707. If all molecular orbitals were filled, then there would have to be one electron in each spin state on each atom, and this gives rise to a second criterion for c-values, namely that the sum of the squares of all the c-values on any one atom in all the molecular orbitals must also equal one. Thus the antibonding orbital of hydrogen, [Pg.7]

Figure 1.7 The radial distribution function, P(r), for the Is atomic orbital in hydrogen. Note the maximum occurs at r = 1 Bohr unit.

As before, attempt to restore the hydrogen match, by substituting the Slater exponent value for the 2s orbital in hydrogen in cells G 3 and G 7. This leads to Figure 1.21. 

Table 1.7 Huzinaga s sto-ng ls> basis sets for tbe Is [Slater] orbital in hydrogen. ...

Exercise 1.11. You should now test your understanding of this first chapter by writing spreadsheets to construct Figures 1.27 and 1.28, which compare Huzinaga s Gaussian sets and the corresponding Slater functions for the Is and 2p orbitals in hydrogen. 

The (4-31) basis set for the Is orbital in hydrogen is an early example of a split-basis set. From Huzinaga s original linear combination of four primitive Gaussians as in Table 1.9,... 

Little modification of fig3-8.xls is required to test the modelling capacity of Is designed basis sets for 2s and higher orbitals in hydrogen and by scaling in other atoms. 

Figure 3.13a Two conditions of the worksheet 2s in fig3-13.xls. Approximations to both the Is and 2s orbitals in hydrogen are made using the same sto-3g) basis of Table 1.6. This leads to the indifferent agreement shown in the first diagram, which is improved dramatically when SOLVER is applied to minimize K 3 with respect to the coefficients and exponents of the basis set in cells C 6 to E 7. The Is worksheet is not shown. Note that the design of the spreadsheet is based on fig3-l l.xls, but now with both coefficients C and Ci redundant and set to 1.0. Note, too, especially, the changed constants, J 7, J 8 and J 9, on each of these spreadsheets.

Figure 4.1 Detail from spreadsheet fig4-l.xls on the CDROM for the calculation of the energy of the Is orbital in hydrogen using the numerical approach. In this particular case, the Slater orbital function is the correct analytical wave function of equation 4.9, but the integration is over the radial coordinate only, so that the normalization constant is the value given in Table 1.1.

For calculations of the energy of p orbitals in hydrogen, the centrifugal term in the Hamiltonian (61), which prevents p and d electrons approaching the nucleus, is greater than zero. Figure 4.3 displays the total atomic potential term (64) for... 

Thus, in a calculation of the energy of the 2p orbital in hydrogen, we must allow for this extra term. Again, in the extra term, there is a denominator, which goes to zero with the radius r . In addition, the spreadsheet design must include the different functional form for the 2p function and the effect of the grad operator. 

The Slater function for the 1 s orbital in hydrogen is the exact eigenfunction of Table 1.1. So the first significant calculation involves the Slater 2s radial function. From equation 1.12,... 

Figure 4.9 Application of the canonical orthonormalization procedure of Section 3.6 to the calculation of the 1 s and 2s eigenfunctions and eigenvalues approximations for the Is and 2s orbitals in hydrogen over Slater functions. Note the exact fit of the Is Slater, which is an eigenfunction of the Fock matrix for the hydrogen atom and the relatively close agreement of the ls/2s linear combinations based on simple canonical orthogonalization and also direct orthonormalization using the matrix procedure of Section 3.7.

Finally, note the extra detail in the chart of Figure 4.9. The linear combinations to form the approximation to the 2s orbital in hydrogen are presented for the choice of canonical orthogonalization based only on the Jacobi diagonalized overlap matrix and for the choice leading to direct orthonormalization, involving the matrix discussed in Section 3.7... 

More significantly, perhaps, little modification is needed to investigate the modelling of 2s and 2p orbitals in hydrogen and the utility of split-basis sets. 

Canonical orthonormallzallon to calculate iKe energies of the 1 s and 2s orbitals in hydrogen ... 

Figure 4.16 The results found for the calculations on the energy of the Is orbital in hydrogen, using Huzinaga s Is basis sets listed in Table 1.7.

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