Comprehensive Study Notes

This section presents some examples of SCF MO wave functions for diatomic molecules.
The spatial orbitals $\phi{i}$ in an MO wave function are each expressed as a linear combination of a set of one-electron basis functions $\chi{s}$ :

\(
\begin{equation}
\phi{i}=\sum{s} c{s i} \chi{s} \tag{13.122}
\end{equation}
\)

For SCF calculations on diatomic molecules, one can use Slater-type orbitals [Eq. (11.14)] centered on the various atoms of the molecule as the basis functions. (For an alternative choice, see Section 15.4.) The procedure used to find the coefficients $c_{s i}$ of the basis functions in each SCF MO is discussed in Section 14.3. To have a complete set of AO basis functions, an infinite number of Slater orbitals are needed, but the true molecular Hartree-Fock wave function can be closely approximated with a reasonably small number of carefully chosen Slater orbitals. A minimal basis set for a molecular SCF calculation consists of a single basis function for each inner-shell AO and each valence-shell AO of each atom. An extended basis set is a set that is larger than a minimal set. Minimal-basis-set SCF calculations are easier than extended-basis-set calculations, but the latter are much more accurate.

SCF wave functions using a minimal basis set were calculated by Ransil for several light diatomic molecules [B. J. Ransil, Rev. Mod. Phys., 32, 245 (1960)]. As an example, the SCF MOs for the ground state of $\mathrm{Li}{2}$ [MO configuration $\left(1 \sigma{g}\right)^{2}\left(1 \sigma{u}\right)^{2}\left(2 \sigma{g}\right)^{2}$ ] at $R=R{e}$ are
$1 \sigma{g}=0.706\left(1 s{a}+1 s{b}\right)+0.009\left(2 s{\perp a}+2 s{\perp b}\right)+0.0003\left(2 p \sigma{a}+2 p \sigma{b}\right)$
$1 \sigma{u}=0.709\left(1 s{a}-1 s{b}\right)+0.021\left(2 s{\perp a}-2 s{\perp b}\right)+0.003\left(2 p \sigma{a}-2 p \sigma{b}\right)$
$2 \sigma{g}=-0.059\left(1 s{a}+1 s{b}\right)+0.523\left(2 s{\perp a}+2 s{\perp b}\right)+0.114\left(2 p \sigma{a}+2 p \sigma{b}\right)$
The AO functions in these equations are STOs, except for $2 s_{\perp}$. A Slater-type $2 s$ AO has no radial nodes and is not orthogonal to a $1 s$ STO. The Hartree-Fock $2 s$ AO has one radial node $(n-l-1=1)$ and is orthogonal to the $1 s$ AO. We can form an orthogonalized $2 s$ orbital with the proper number of nodes by taking the following normalized linear combination of $1 s$ and $2 s$ STOs of the same atom (Schmidt orthogonalization):

\(
\begin{equation}
2 s_{\perp}=\left(1-S^{2}\right)^{-1 / 2}(2 s-S \cdot 1 s) \tag{13.124}
\end{equation}
\)

where $S$ is the overlap integral $\langle 1 s \mid 2 s\rangle$. Ransil expressed the $\mathrm{Li}{2}$ orbitals using the (nonorthogonal) $2 s$ STO, but since the orthogonalized $2 s{\perp}$ function gives a better representation of the $2 s \mathrm{AO}$, the orbitals have been rewritten using $2 s{\perp}$. This changes the $1 s$ and $2 s$ coefficients, but the actual orbital is, of course, unchanged; see Prob. 13.37. The notation $2 p \sigma$ for an AO indicates that the $p$ orbital points along the molecular $(z)$ axis; that is, a $2 p \sigma$ AO is a $2 p{z} \mathrm{AO}$. (The $2 p{x}$ and $2 p{y} \mathrm{AOs}$ are called $2 p \pi$ AOs.) The optimum orbital exponents for the orbitals in (13.123) are $\zeta{1 s}=2.689, \zeta{2 s}=0.634, \zeta_{2 p \sigma}=0.761$.

Our previous simple expressions for these MOs were

\(
\begin{aligned}
& 1 \sigma{g}=\sigma{g} 1 s=2^{-1 / 2}\left(1 s{a}+1 s{b}\right) \
& 1 \sigma{u}=\sigma{u}^{*} 1 s=2^{-1 / 2}\left(1 s{a}-1 s{b}\right) \
& 2 \sigma{g}=\sigma{g} 2 s=2^{-1 / 2}\left(2 s{a}+2 s{b}\right)
\end{aligned}
\)

Comparison of these with (13.123) shows the simple LCAO functions to be reasonable first approximations to the minimal-basis-set SCF MOs. The approximation is best for the $1 \sigma{g}$ and $1 \sigma{u} \mathrm{MOs}$, whereas the $2 \sigma_{g} \mathrm{MO}$ has substantial $2 p \sigma \mathrm{AO}$ contributions in addition to the $2 s$ AO contributions. For this reason the notation of the third column of Table 13.1 (Section 13.7) is preferable to the separated-atoms MO notation. The substantial amount of $2 s-2 p \sigma$ hybridization is to be expected, since the $2 s$ and $2 p$ AOs are close in energy [see Eq. (9.27)]. The hybridization allows for the polarization of the $2 s \mathrm{AOs}$ in forming the molecule.

Let us compare the $3 \sigma{g} \mathrm{MO}$ of the $\mathrm{F}{2}$ ground state at $R_{e}$ as calculated by Ransil using a minimal basis set with that calculated by Wahl using an extended basis set [A. C. Wahl, J. Chem. Phys., 41, 2600 (1964)]:

\(
\begin{gathered}
3 \sigma{g, \text { min }}=0.038\left(1 s{a}+1 s{b}\right)-0.184\left(2 s{a}+2 s{b}\right)+0.648\left(2 p \sigma{a}+2 p \sigma{b}\right) \
\zeta{1 s}=8.65, \quad \zeta{2 s}=2.58, \quad \zeta{2 p \sigma}=2.49 \
3 \sigma{g, \mathrm{ext}}=0.048\left(1 s{a}+1 s{b}\right)+0.003\left(1 s{a}^{\prime}+1 s{b}^{\prime}\right)-0.257\left(2 s{a}+2 s{b}\right) \
+0.582\left(2 p \sigma{a}+2 p \sigma{b}\right)+0.307\left(2 p \sigma{a}^{\prime}+2 p \sigma{b}^{\prime}\right)+0.085\left(2 p \sigma{a}^{\prime \prime}+2 p \sigma{b}^{\prime \prime}\right) \
-0.056\left(3 s{a}+3 s{b}\right)+0.046\left(3 d \sigma{a}+3 d \sigma{b}\right)+0.014\left(4 f \sigma{a}+4 f \sigma{b}\right) \
\zeta{1 s}=8.27, \quad \zeta{1 s^{\prime}}=13.17, \quad \zeta{2 s}=2.26 \
\zeta{2 p \sigma}=1.85, \quad \zeta{2 p \sigma^{\prime}}=3.27, \quad \zeta{2 p \sigma^{\prime \prime}}=5.86 \
\zeta{3 s}=4.91, \quad \zeta{3 d \sigma}=2.44, \quad \zeta{4 f \sigma}=2.83
\end{gathered}
\)

Just as several STOs are needed to give an accurate representation of Hartree-Fock AOs (Section 11.1), one needs more than one STO of a given $n$ and $l$ in the linear combination of STOs that is to accurately represent the Hartree-Fock MO. The primed and double-primed AOs in the extended-basis-set function are STOs with different orbital exponents. The $3 d \sigma$ and $4 f \sigma$ AOs are AOs with quantum number $m=0$, that is, the $3 d{0}$ and $4 f{0}$ AOs. The total energies found are $-197.877 E{\mathrm{h}}$ and $-198.768 E{\mathrm{h}}$ for the minimal and extended calculations, respectively (where $E{\mathrm{h}}$ is the hartree). Extrapolation of calculations using much larger basis sets than Wahl used gives the Hartree-Fock $\mathrm{F}{2}$ energy at $R{e}$ as $-198.773 E{\mathrm{h}}$ [L. Bytautas et al., J. Chem. Phys., 127, 164317 (2007)]. The experimental energy of $\mathrm{F}{2}$ at $R{e}$ is $U\left(R{e}\right)=-199.672 E{\mathrm{h}}$. The correlation-energy definition (11.16) uses the nonrelativistic energy of the molecule. The relativistic contribution to the $\mathrm{F}{2}$ energy has been calculated to be $-0.142 E{\mathrm{h}}$, so the exact nonrelativistic $\mathrm{F}{2}$ energy at $R{e}$ is $-199.672 E{\mathrm{h}}+0.142 E{\mathrm{h}}=-199.530 E{\mathrm{h}}$. Therefore, the correlation energy in $\mathrm{F}{2}$ is $-199.530 E{\mathrm{h}}+198.773 E{\mathrm{h}}=-0.757 E_{\mathrm{h}}=-20.6 \mathrm{eV}$.

FIGURE 13.20 Hartree-Fock MO electron-density contours for the ground electronic state of $\mathrm{Li}_{2}$ as calculated by Wahl. [A. C. Wahl, Science,
151, 961 (1966); Scientific
American, April 1970, p. 54;
Atomic and Molecular Structure: 4 Wall Charts, McGraw-Hill, 1970.]

In discussing $\mathrm{H}{2}^{+}$and $\mathrm{H}{2}$, we saw how hybridization (the mixing of different AOs of the same atom) improves molecular wave functions. There is a tendency to think of hybridization as occurring only for certain molecular geometries. The SCF calculations make clear that all MOs are hybridized to some extent. Thus any diatomicmolecule $\sigma \mathrm{MO}$ is a linear combination of $1 s, 2 s, 2 p{0}, 3 s, 3 p{0}, 3 d_{0}, \ldots$ AOs of the separated atoms.

To aid in deciding which AOs contribute to a given diatomic MO, we use two rules. First, only $\sigma$-type $\mathrm{AOs}(s, p \sigma, d \sigma, \ldots)$ can contribute to a $\sigma \mathrm{MO}$; only $\pi$-type AOs ( $p \pi, d \pi, \ldots$ ) can contribute to a $\pi \mathrm{MO}$; and so on. Second, only AOs of reasonably similar energy contribute substantially to a given MO. (For examples, see the minimal- and extended-basis-set MOs quoted above.)

Wahl plotted the contours of the near Hartree-Fock molecular orbitals of homonuclear diatomic molecules from $\mathrm{H}{2}$ through $\mathrm{F}{2}$. Figure $\mathbf{1 3 . 2 0}$ shows these plots for $\mathrm{Li}_{2}$.

Of course, Hartree-Fock wave functions are only approximations to the true wave functions. It is possible to prove that a Hartree-Fock wave function gives a very good approximation to the electron probability density $\rho(x, y, z)$ for nuclear configurations in the region of the equilibrium configuration. A molecular property that involves only one-electron operators can be expressed as an integral involving $\rho$; see Eq. (14.8). Consequently, such properties are accurately calculated using Hartree-Fock wave functions. An example is the molecular dipole moment [Eq. (14.21)]. For example, the LiH dipole moment calculated with a near Hartree-Fock $\psi$ is 6.00 D (debyes) [S. Green, J. Chem. Phys., 54, 827 (1971)], compared with the experimental value 5.83 D. (One debye $=3.33564 \times 10^{-30} \mathrm{C}$ m.) For NaCl , the calculated and experimental dipole moments are 9.18 D and 9.02 D [R. L. Matcha, J. Chem. Phys., 48, 335 (1968)]. An error of about 0.2 D is typical in such calculations, but where the dipole moment is small, the percent error can be large. An extreme example is CO, for which the experimental moment is 0.11 D with the polarity $\mathrm{C}^{-} \mathrm{O}^{+}$, but the near-Hartree-Fock moment is 0.27 D with the wrong polarity $\mathrm{C}^{+} \mathrm{O}^{-}$. However, a configuration-interaction wave function gives 0.12 D with the correct polarity [S. Green, J. Chem. Phys., 54, 827 (1971)].

A major weakness of the Hartree-Fock method is its failure to give accurate molecular dissociation energies. For example, an extended-basis-set calculation [P. E. Cade et al., J. Chem. Phys., 44, 1973 (1966)] gives $D{e}=5.3 \mathrm{eV}$ for $\mathrm{N}{2}$, as compared with the true value 9.9 eV . (To calculate the Hartree-Fock $D{e}$, the molecular energy at the minimum in the $U(R)$ Hartree-Fock curve is subtracted from the sum of the Hartree-Fock energies of the separated atoms.) A related defect of Hartree-Fock molecular wave functions is that the energy approaches the wrong limit as $R \rightarrow \infty$. Recall the MO discussion of $\mathrm{H}{2}$.

13.15 MO Treatment of Heteronuclear Diatomic Molecules

The treatment of heteronuclear diatomic molecules is similar to that for homonuclear diatomic molecules. We first consider the MO description.

Suppose the two atoms have atomic numbers that differ only slightly; an example is CO . We could consider CO as being formed from the isoelectronic molecule $\mathrm{N}{2}$ by a gradual transfer of charge from one nucleus to the other. During this hypothetical transfer, the original $\mathrm{N}{2}$ MOs would slowly vary to give finally the CO MOs. We therefore expect the CO molecular orbitals to resemble somewhat those of $\mathrm{N}{2}$. For a heteronuclear diatomic molecule such as CO, the symbols used for the MOs are similar to those for homonuclear diatomics. However, for a heteronuclear diatomic, the electronic Hamiltonian (13.5) is not invariant with respect to inversion of the electronic coordinates (that is, $\hat{H}{\text {el }}$ does not commute with $\hat{\Pi}$ ), and the $g$, u property of the MOs disappears. The correlation between the $\mathrm{N}_{2}$ and CO shell designations is

MOs of the same symmetry are numbered in order of increasing energy. Because of the absence of the $g, u$ property, the numbers of corresponding homonuclear and heteronuclear MOs differ. Figure 13.21 is a sketch of a contour of the CO $1 \pi{ \pm 1}$ MOs as determined by an extended-basis-set SCF calculation [W. M. Huo, J. Chem. Phys., 43, 624 (1965)]. Note its resemblance to the contour of Fig. 13.13, which is for the $1 \pi{u, \pm 1} \mathrm{MOs}$ of a homonuclear diatomic molecule.

The ground-state configuration of CO is $1 \sigma^{2} 2 \sigma^{2} 3 \sigma^{2} 4 \sigma^{2} 1 \pi^{4} 5 \sigma^{2}$, as compared with the $\mathrm{N}{2}$ configuration $\left(1 \sigma{g}\right)^{2}\left(1 \sigma{u}\right)^{2}\left(2 \sigma{g}\right)^{2}\left(2 \sigma{u}\right)^{2}\left(1 \pi{u}\right)^{4}\left(3 \sigma_{g}\right)^{2}$.

As in homonuclear diatomics, the heteronuclear diatomic MOs are approximated as linear combinations of atomic orbitals. The coefficients are found by the procedure of Section 14.3. For example, a minimal-basis-set SCF calculation using Slater AOs (with nonoptimized exponents given by Slater's rules) gives for the CO $5 \sigma, 1 \pi$, and $2 \pi \mathrm{MOs}$ at $R=R_{e}$ [B. J. Ransil, Rev. Mod. Phys., 32, 245 (1960)]:

\(
\begin{gathered}
5 \sigma=0.027\left(1 s{\mathrm{C}}\right)+0.011\left(1 s{\mathrm{O}}\right)+0.739\left(2 s{\perp \mathrm{C}}\right)+0.036\left(2 s{\perp \mathrm{O}}\right) \
-0.566\left(2 p \sigma{\mathrm{C}}\right)-0.438\left(2 p \sigma{\mathrm{O}}\right) \
1 \pi=0.469\left(2 p \pi{\mathrm{C}}\right)+0.771\left(2 p \pi{\mathrm{O}}\right), \quad 2 \pi=0.922\left(2 p \pi{\mathrm{C}}\right)-0.690\left(2 p \pi{\mathrm{O}}\right)
\end{gathered}
\)

The expressions for the $\pi$ MOs are simpler than those for the $\sigma$ MOs because $s$ and $p \sigma$ AOs cannot contribute to $\pi$ MOs. For comparison, the corresponding MOs in $\mathrm{N}{2}$ at $R=R{e}$ are (Ransil, op. cit.):

\(
\begin{gathered}
3 \sigma{g}=0.030\left(1 s{a}+1 s{b}\right)+0.395\left(2 s{\perp a}+2 s{\perp b}\right)-0.603\left(2 p \sigma{a}+2 p \sigma{b}\right) \
1 \pi{u}=0.624\left(2 p \pi{a}+2 p \pi{b}\right), \quad 1 \pi{g}=0.835\left(2 p \pi{a}-2 p \pi_{b}\right)
\end{gathered}
\)

The resemblance of CO and $\mathrm{N}_{2} \mathrm{MOs}$ is apparent. The $1 \sigma \mathrm{MO}$ in CO is found to be nearly the same as a $1 s$ oxygen-atom AO ; the $2 \sigma \mathrm{MO}$ in CO is essentially a carbon-atom 1s AO.

In general, for a heteronuclear diatomic molecule $A B$ where the valence $A O$ s of each atom are of $s$ and $p$ type and where the valence AOs of A do not differ greatly in energy from the valence AOs of B, we can expect the Fig. 13.17 pattern of

\(
\sigma s<\sigma^{} s<\pi p<\sigma p<\pi^{} p<\sigma^{*} p
\)

Figure 13.21 Cross section of a contour of the $1 \pi_{ \pm 1}$ MOs in CO .

valence-shell MOs formed from $s$ and $p$ valence-shell AOs to hold reasonably well. Figure 13.17 would be modified in that each valence AO of the more electronegative atom would lie below the corresponding valence AO of the other atom.

When the valence-shell AO energies of B lie very substantially below those of A , the $s$ and $p \sigma$ valence AOs of B lie below the $s$ valence-shell AO of A , and this affects which AOs contribute to each MO. Consider the molecule BF, for example. A minimal-basis-set calculation [Ransil, Rev. Mod. Phys., 32, 245 (1960)] gives the $1 \sigma \mathrm{MO}$ as essentially $1 s{\mathrm{F}}$ and the $2 \sigma \mathrm{MO}$ as essentially $1 s{\mathrm{B}}$. The $3 \sigma \mathrm{MO}$ is predominantly $2 s{\mathrm{F}}$, with small amounts of $2 s{\mathrm{B}}, 2 p \sigma{\mathrm{B}}$, and $2 p \sigma{\mathrm{F}}$. The $4 \sigma$ MO is predominantly $2 p \sigma{\mathrm{F}}$, with significant amounts of $2 s{\mathrm{B}}$ and $2 s{\mathrm{F}}$ and a small amount of $2 p \sigma{\mathrm{B}}$. This is quite different from $\mathrm{N}{2}$, where the corresponding MO is formed predominantly from the $2 s$ AOs on each N . The $1 \pi$ MO is a bonding combination of $2 p \pi{\mathrm{B}}$ and $2 p \pi{\mathrm{F}}$. The $5 \sigma$ MO is predominantly $2 s{\mathrm{B}}$, with a substantial contribution from $2 p \sigma{\mathrm{B}}$ and a significant contribution from $2 p \sigma{\mathrm{F}}$. This is unlike the corresponding MO in $\mathrm{N}{2}$, where the largest contributions are from $2 p \sigma$ MOs on each atom. The $2 \pi \mathrm{MO}$ is an antibonding combination of $2 p \pi{\mathrm{B}}$ and $2 p \pi{\mathrm{F}}$. The $6 \sigma$ MO has important contributions from $2 p \sigma{\mathrm{B}}, 2 s{\mathrm{B}}, 2 s{\mathrm{F}}$, and $2 p \sigma_{\mathrm{F}}$.

We see from Fig. 11.2 that the $2 p{\mathrm{F}}$ AO lies well below the $2 s{\mathrm{B}} \mathrm{AO}$. This causes the $2 p \sigma{\mathrm{F}} \mathrm{AO}$ to contribute substantially to lower-lying MOs and the $2 s{\mathrm{B}} \mathrm{AO}$ to contribute substantially to higher-lying MOs, as compared with what happens in $\mathrm{N}{2}$. (This effect occurs in CO, although to a lesser extent. Note the very substantial contribution of $2 s{\mathrm{C}}$ to the $5 \sigma \mathrm{MO}$. Also, the $4 \sigma \mathrm{MO}$ in CO has a very substantial contribution from $2 p \sigma_{\mathrm{O}}$.)

For a diatomic molecule AB where each atom has $s$ and $p$ valence-shell AOs (this excludes H and transition elements) and where the A and B valence AOs differ widely in energy, we may expect the pattern of valence MOs to be $\sigma<\sigma<\pi<\sigma<\pi<\sigma$, but it is not so easy to guess which AOs contribute to the various MOs or the bonding or antibonding character of the MOs. By feeding the valence electrons into these MOs, we can make a plausible guess as to the number of unpaired electrons and the ground term of the AB molecule (Prob. 13.38).

Diatomic hydrides are a special case, since H has only a $1 s$ valence AO. Consider HF as an example. The ground-state configurations of the atoms are $1 s$ for H and $1 s^{2} 2 s^{2} 2 p^{5}$ for F . We expect the filled $1 s$ and $2 s \mathrm{~F}$ subshells to take little part in the bonding. The four $2 p \pi$ fluorine electrons are nonbonding (there are no $\pi$ valence AOs on H ). The hydrogen $1 s \mathrm{AO}$ and the fluorine $2 p \sigma \mathrm{AO}$ have the same symmetry $(\sigma)$ and have rather similar energies (Fig. 11.2), and a linear combination of these two AOs will form a $\sigma$ MO for the bonding electron pair:

\(
\phi=c{1}\left(1 s{\mathrm{H}}\right)+c{2}\left(2 p \sigma{\mathrm{F}}\right)
\)

where the contributions of $1 s{\mathrm{F}}$ and $2 s{\mathrm{F}}$ to this MO have been neglected. Since F is more electronegative than H , we expect that $c{2}>c{1}$. (In addition, the $1 s{\mathrm{H}}$ and $2 p \sigma{\mathrm{F}} \mathrm{AOs}$ form an antibonding MO, which is unoccupied in the ground state.)

The picture of HF just given is only a crude qualitative approximation. A minimal-basis-set SCF calculation using Slater orbitals with optimized exponents gives as the MOs of HF [B. J. Ransil, Rev. Mod. Phys., 32, 245 (1960)]

\(
\begin{gathered}
1 \sigma=1.000\left(1 s{\mathrm{F}}\right)+0.012\left(2 s{\perp \mathrm{F}}\right)+0.002\left(2 p \sigma{\mathrm{F}}\right)-0.003\left(1 s{\mathrm{H}}\right) \
2 \sigma=-0.018\left(1 s{\mathrm{F}}\right)+0.914\left(2 s{\perp \mathrm{F}}\right)+0.090\left(2 p \sigma{\mathrm{F}}\right)+0.154\left(1 s{\mathrm{H}}\right) \
3 \sigma=-0.023\left(1 s{\mathrm{F}}\right)-0.411\left(2 s{\perp \mathrm{F}}\right)+0.711\left(2 p \sigma{\mathrm{F}}\right)+0.516\left(1 s{\mathrm{H}}\right) \
1 \pi{+1}=\left(2 p \pi{+1}\right){\mathrm{F}}, \quad 1 \pi{-1}=\left(2 p \pi{-1}\right){\mathrm{F}}
\end{gathered}
\)

The ground-state MO configuration of HF is $1 \sigma^{2} 2 \sigma^{2} 3 \sigma^{2} 1 \pi^{4}$. The $1 \sigma$ MO is virtually identical with the $1 s$ fluorine AO. The $2 \sigma$ MO is pretty close to the $2 s$ fluorine AO. The $1 \pi$ MOs are required by symmetry to be the same as the corresponding fluorine $\pi$ AOs. The bonding $3 \sigma$ MO has its largest contribution from the $2 p \sigma$ fluorine and $1 s$ hydrogen AOs, as would be expected from the discussion of the preceding paragraph. However, the $2 s$ fluorine AO makes a substantial contribution to this MO. (Since a single $2 s$ function is only an approximation to the $2 s \mathrm{AO}$ of F , we cannot use this calculation to say exactly how much $2 s \mathrm{AO}$ character the $3 \sigma \mathrm{HF}$ molecular orbital has.)

For qualitative discussion (but not quantitative work), it is useful to have simple approximations for heteronuclear diatomic MOs. In the crudest approximation, we can take each valence MO of a heteronuclear diatomic molecule as a linear combination of two AOs $\phi{a}$ and $\phi{b}$, one on each atom. (As the discussions of CO and BF show, this approximation is often quite inaccurate.) From the two AOs, we can form two MOs:

\(
c{1} \phi{a}+c{2} \phi{b} \text { and } c{1}^{\prime} \phi{a}+c{2}^{\prime} \phi{b}
\)

The lack of symmetry in the heteronuclear diatomic makes the coefficients unequal in magnitude. The coefficients are determined by solving the secular equation [see Eq. (13.45)]

\(
\begin{gather}
\left|\begin{array}{cc}
H{a a}-W & H{a b}-W S{a b} \
H{a b}-W S{a b} & H{b b}-W
\end{array}\right|=0 \
\left(H{a a}-W\right)\left(H{b b}-W\right)-\left(H{a b}-W S{a b}\right)^{2}=0 \tag{13.125}
\end{gather}
\)

where $\hat{H}$ is some sort of effective one-electron Hamiltonian. Suppose that $H{a a}>H{b b}$, and let $f(W)$ be defined as the left side of (13.125). The overlap integral $S{a b}$ is less than 1 (except at $R=0$ ). [A rigorous proof of this follows from Eq. (3-114) in Margenau and Murphy.] The coefficient of $W^{2}$ in $f(W)$ is $\left(1-S{a b}^{2}\right)>0$; therefore $f(\infty)=f(-\infty)=+\infty>0$. For $W=H{a a}$ or $H{b b}$, the first product in (13.125) vanishes. Hence $f\left(H{a a}\right)<0$ and $f\left(H{b b}\right)<0$. The roots of (13.125) occur where $f(W)$ equals 0 . Hence, by continuity, one root must be between $+\infty$ and $H{a a}$ and the other between $H{b b}$ and $-\infty$. Therefore, the orbital energy of one MO is less than both $H{a a}$ and $H{b b}$ (the energies of the two AOs in the molecule; Section 13.5), and the energy of the other MO is greater than both $H{a a}$ and $H{b b}$. One bonding and one antibonding MO are formed from the two AOs. Figure 13.22 shows the formation of bonding and antibonding MOs from two AOs, for the homonuclear and heteronuclear cases. These figures are gross oversimplifications, since a given MO has contributions from many AOs, not just two.

The coefficients $c{1}$ and $c{2}$ in the bonding heteronuclear MO in Fig. 13.22 are both positive, so as to build up charge between the nuclei. For the antibonding heteronuclear MO, the coefficients of $\phi{a}$ and $\phi{b}$ have opposite signs, causing charge depletion between the nuclei.

FIGURE 13.22 Formation of bonding and antibonding MOs from AOs in the homonuclear and heteronuclear cases. (See Prob. 13.39.)

13.16 VB Treatment of Heteronuclear Diatomic Molecules

Consider the valence-bond ground state wave function of HF. We expect a single bond to be formed by the pairing of the hydrogen $1 s$ electron and the unpaired fluorine $2 p \sigma$ electron. The Heitler-London function corresponding to this pairing is [Eq. (13.117)]

\(
\begin{equation}
\phi{\text {cov }}=\left|\cdots 1 s{\mathrm{H}} \overline{2 p \sigma{\mathrm{F}}}\right|-\left|\cdots \overline{1 s{\mathrm{H}}} 2 p \sigma_{\mathrm{F}}\right| \tag{13.126}
\end{equation}
\)

where the dots stand for $1 s{\mathrm{F}} \overline{1 s{\mathrm{F}}} 2 s{\mathrm{F}} \overline{2 s{\mathrm{F}}} 2 p \pi{x \mathrm{~F}} \overline{2 p \pi{x \mathrm{~F}}} 2 p \pi{y \mathrm{~F}} \overline{2 p \pi{y \mathrm{~F}}}$. This function is essentially covalent, the electrons being shared by the two atoms. However, the high electronegativity of fluorine leads us to include a contribution from an ionic structure as well. An ionic valence-bond function has the form $\phi{a}(1) \phi{a}(2)$ [Eq. (13.109)]. Introduction of the required antisymmetric spin factor gives as the valence-bond function for an ionic structure in HF :

\(
\phi{\text {ion }}=\left|\cdots 2 p \sigma{\mathrm{F}} \overline{2 p \sigma_{\mathrm{F}}}\right|
\)

The VB wave function is then written as

\(
\begin{equation}
\phi=c{1} \phi{\mathrm{cov}}+c{2} \phi{\mathrm{ion}} \tag{13.127}
\end{equation}
\)

The optimum values of $c{1}$ and $c{2}$ are found by the variation method. This leads to the usual secular equation. We have ionic-covalent "resonance," involving the structures H-F and $\mathrm{H}^{+} \mathrm{F}^{-}$. The true molecular structure is intermediate between the covalent and ionic structures. A term $c{3}\left|1 s{\mathrm{H}} \overline{s_{\mathrm{H}}}\right|$ corresponding to the ionic structure $\mathrm{H}^{-} \mathrm{F}^{+}$could also be included in the wave function, but this should contribute only slightly for HF. For molecules that are less ionic, both ionic structures might well be included.

For a highly ionic molecule such as NaCl , we expect the VB function to have $c{2} \gg c{1}$. It might be thought that NaCl would dissociate to $\mathrm{Na}^{+}$and $\mathrm{Cl}^{-}$ions, but this is not true. The ionization energy of Na is 5.1 eV , while the electron affinity of Cl is only 3.6 eV . Hence, in the gas phase the neutral separated ground-state atoms $\mathrm{Na}+\mathrm{Cl}$ are more stable than the ground-state separated ions $\mathrm{Na}^{+}+\mathrm{Cl}^{-}$. (In aqueous solution the ions are more stable because of the hydration energy, which makes the separated ions more stable than even the diatomic NaCl molecule.) If the nuclei are slowly pulled apart, a gas-phase NaCl molecule will dissociate to neutral atoms. Therefore, as $R$ increases from $R{e}$, the ratio $c{2} / c_{1}$ in (13.127) must decrease, becoming zero at $R=\infty$. For intermediate values of $R$, the Coulombic attraction between the ions is greater than the $1.5-\mathrm{eV}$ difference between the ionization potential and electron affinity, and the molecule is largely ionic. For very large $R$, the Coulombic attraction between the ions is less than 1.5 eV , and the molecule is largely covalent. However, if the nuclei in NaCl are pulled apart very rapidly, then the electrons will not have a chance to adjust their wave function from the ionic to the covalent wave function, and both bonding electrons will go with the chlorine nucleus, giving dissociation into ions.

Cesium has the lowest ionization energy, 3.9 eV . Chlorine has the highest electron affinity, 3.6 eV . Thus, even for CsCl and CsF , the separated ground-state neutral atoms are more stable than the separated ground-state ions. There are, however, cases of excited states of diatomic molecules that dissociate to ions.

13.17 The Valence-Electron Approximation

Suppose we want to treat $\mathrm{Cs}_{2}$, which has 110 electrons. In the MO method, we would start by writing down a $110 \times 110$ Slater determinant of molecular orbitals. We would then approximate the MOs by functions containing variational parameters and go on
to minimize the variational integral. Clearly, the large number of electrons makes this a formidable task. One way to simplify the problem is to divide the electrons into two groups: the 108 core electrons and the two $6 s$ valence electrons, which provide bonding. We then try to treat the valence electrons separately from the core, taking the molecular energy as the sum of core- and valence-electron energies. This approach, introduced in the 1930s, is called the valence-electron approximation.

The simplest approach is to regard the core electrons as point charges coinciding with the nucleus. For $\mathrm{Cs}{2}$ this would give a Hamiltonian for the two valence electrons that is identical with the electronic Hamiltonian for $\mathrm{H}{2}$. If we then go ahead and minimize the variational integral for the valence electrons in $\mathrm{Cs}{2}$, with no restrictions on the valence-electron trial functions, we will clearly be in trouble. Such a procedure will cause the valence-electrons' MO to "collapse" to the $\sigma{g} 1 s$ MO, since the core electrons are considered absent. To avoid this collapse, one can impose the constraint that the variational functions used for the valence electrons be orthogonal to the orbitals of the core electrons. Of course, the task of keeping the valence orbitals orthogonal to the core orbitals means more work. A somewhat different approach is to drop the approximation of treating the core electrons as coinciding with the nucleus, and to treat them as a charge distribution that provides some sort of effective repulsive potential for the motion of the valence electrons. This leads to an effective Hamiltonian for the valence electrons, which is then used in the variational integral. The valence-electron approximation is widely used in approximate treatments of polyatomic molecules (Chapter 17).