Molecular Theory of the Chemical Bond

Yu. B. Rumer, Moscow

In recent years, chiefly in the works of Hund, Herzberg, and Mulliken, a new direction has arisen in the quantum theory of the chemical bond, somewhat different from the direction that originated in the work of Heitler–London. According to Heitler–London, the chemical bond is realized as a result of the interaction of the spin moments of the reacting atoms. According to this theory, the formation of a homopolar bond requires the participation of at least two electrons interacting through their spins. In this respect the Heitler–London theory is a quantum model for Lewis’s electron pairs. Characteristic of this theory is the circumstance that, in the molecule, the participating atoms preserve their individuality. The whole theory answers the question of the tendency of atoms to combine into a molecule if they are brought together from infinite separation.

In contrast to this point of view, the theory set forth here proceeds from considering a molecule as a system consisting of several positively charged centers, in whose field negative electrons move. The energy of such a system will depend on the mutual distances between the nuclei. The question arises of the stability of such a system, and the answer to it immediately yields a number of rules similar to those obtained empirically by chemists.

Let us turn to diatomic molecules. Let us fix two nuclei with charges \(Z_1\) and \(Z_2\) at a distance \(R_{12}\) from one another and begin to fill, one after another, the molecular orbits of these two centers. We shall have a construction principle (Aufbauprinzip) for the molecule, analogous to the construction principle used in atomic theory. It should be noted that after each new electron we shall have to give the nuclei the possibility of taking a new equilibrium position corresponding to the new electronic configuration of the molecule. In atoms, naturally, we did not have to change the character of the field after each new electron. Electrons, moving along their molecular orbits, experience a strong interaction with one another. But, just as is done in atomic theory, we shall neglect this interaction and assume that each electron moves independently of the others in the field of two nuclei. In this case the potential curve of the energy of the molecule will be composed of the potential energy curves of the individual electrons. It will turn out that

some orbitals, when nuclei approach each other, increase their energy. Such orbitals, obviously, will oppose the approach of atoms. Conversely, other orbitals, when nuclei approach each other, will decrease their energy and thereby promote the chemical bond.

Following Herzberg, we shall call the first orbitals loosening (lockernd), and the second—binding (bindend). The question of the stability of a diatomic molecule reduces to the question of the final energy balance of all the orbitals of the molecule.

Just as the theory of the atom is begun with the problem of hydrogen and helium, we shall begin the exposition of the theory of diatomic molecules with the simplest molecules: the hydrogen molecular ion \(H_2^+\) and the hydrogen molecule \(H_2\). Then we shall pass to the construction of the periodic system for diatomic molecules, and then touch upon polyatomic molecules, whose theory is only now beginning to be developed.

We shall use atomic units of measurement throughout. Let the electron charge \(e=1\), \(h/2\pi=1\), and the electron mass \(m=1\). Then the unit of length will be the Bohr radius \(a_0=0.553\,\text{\AA}\), and the unit of energy will be twice the ionization work of hydrogen, \(26.08\,V\). We shall measure energy in rydbergs: \(2Ry=1\) atomic unit of energy.

1. THE HYDROGEN MOLECULAR ION

The fundamental problem, analogous to the problem of the hydrogen atom in atomic theory, is, for the problem of diatomic molecules, the problem of the hydrogen molecular ion. We are dealing with the motion of one electron in the field of two protons located at a distance \(R\) from one another.

The Schrödinger equation, for our case, will be:

\[ \left\{-\frac{1}{2}\Delta\psi-\frac{1}{r_a}-\frac{1}{r_b}\right\}\psi=E\psi. \tag{1.1} \]

Transition to elliptic coordinates

For the two-center problem the most natural coordinate system is elliptic coordinates:

\[ \xi=\frac{r_a+r_b}{R},\quad \eta=\frac{r_a-r_b}{R},\quad \varphi=\operatorname{arc\,tg}\frac{x}{y}, \]

\[ 1<\xi<\infty,\qquad -1<\eta<+1. \]

In elliptic coordinates the equation will have the form:

\[ \frac{\partial}{\partial \xi}\left\{(\xi^2-1)\frac{\partial \psi}{\partial \xi}\right\} +\frac{\partial}{\partial \eta}\left\{(1-\eta^2)\frac{\partial \psi}{\partial \eta}\right\} +\left\{\frac{1}{\xi^2-1}+\frac{1}{1-\eta^2}\right\}\frac{\partial^2\psi}{\partial\varphi^2} +\frac{R^2}{2}\left\{E(\xi^2-\eta^2)+\frac{4}{R}\xi\right\}\psi=0, \tag{1.2} \]

allowing separation of variables. Put:

\[ \psi(\xi,\eta,\varphi)=X(\xi)Y(\eta)e^{\lambda\varphi}, \tag{1.3} \]

where, by the requirement that the function \(\psi\) be single-valued, \(\lambda\) is an integer.

For \(X(\xi)\) and \(Y(\eta)\) we obtain two equations:

\[ \frac{d}{d\xi}\left\{(\xi^2-1)\frac{dx}{d\xi}\right\} +\left(\frac{1}{2}ER^2\xi^2+2R\xi+A-\frac{m^2}{\xi^2-1}\right)X=0, \tag{1.4} \]

\[ \frac{d}{d\eta}\left\{(\eta^2-1)\frac{dy}{d\eta}\right\} +\left(\frac{1}{2}ER^2+A-\frac{m^2}{\eta^2-1}\right)y=0. \tag{1.5} \]

\[ A\text{ is a function of the parameter }R. \]

Let us note that in our case the energy of the system will, as a parameter, depend on \(R\). Our task will be to find such \(E(R)\) for which equation (1.1) has a finite, single-valued, and continuous solution throughout all space.

In addition, we have deliberately struck out one term from the Schrödinger equation—the interaction of the protons. The point is that, in studying our equation, we shall move the nuclei apart and bring them closer together. We shall have a gain or loss of energy. In doing so it is convenient to exclude the Coulomb energy of the interaction of the protons. In this case we can, at the cost of a finite expenditure of energy, carry out the transformation of the molecule \(H_2^+\) into an ionized helium atom and compare the state of the molecule with the familiar states of the atom. If, however, we need to know the potential curve for the molecule, then instead of the function \(E(R)\) we shall have to consider the function:

\[ E(R)+\frac{1}{R}\quad(\text{electron energy}+\text{energy of interaction of the nuclei}), \]

which already describes the total energy of the molecule.

For what follows we shall introduce the spectroscopic notation accepted in the theory of molecules. The state of the electron is described by the \(\psi\)-function:

\[ X(\xi)Y(\eta)e^{\lambda\varphi}\quad(\lambda\text{ an integer}). \tag{1.3} \]

In this state the component of the angular momentum along the \(z\)-axis, to which in quantum mechanics there corresponds the operator \(-i\,\dfrac{\partial}{\partial\varphi}\), has the definite value \(\lambda\).

Since \(\lambda^2\) enters the equation, the energy value does not depend on the sign of \(\lambda\). Therefore the state corresponding to \(\pm\lambda\) is doubly degenerate. States with values \(\lambda=0,1,2,\ldots\) are called \(\sigma\)-, \(\pi\)-, \(\delta\)-, \(\ldots\) states of the electron in the field of two centers. If the molecule is not at rest, but also undergoes rotation, the degeneracy

disappears and the corresponding term splits. The degree of splitting (the so-called \(\lambda\)-doubling) obviously depends on the speed of rotation.

Investigation of the solution

Let us consider the particular cases of our equations (1.4) and (1.5) for \(R=0\) and \(R=\infty\).

1) \(R=0\). The elliptic coordinates are expressed in terms of spherical ones:

\[ \xi \to \frac{2r}{R},\quad \eta \to \cos \vartheta,\quad \varphi=\varphi, \]

\[ \frac{R\xi}{2}=u,\quad \eta=v,\quad \varphi=\varphi, \]

\[ \varepsilon=\sqrt{-2E}. \]

The equation becomes

\[ \frac{d}{du}\left(u^2\frac{dX}{du}\right)+(2Eu^2+4u+A)X=0, \tag{1.4}\_0 \]

where \(A=-l(l+1)\)

\[ \frac{d}{dv}\left\{(v^2-1)\frac{dY}{dv}\right\}+\left\{A-\frac{\lambda^2}{v^2-1}\right\}Y=0. \tag{1.5}\_0 \]

The problem becomes the problem of ionized helium, whose solutions are known (for negative values of \(E\)):

\[ X(u)=e^{-\varepsilon u}u^l L_{u+1}^{2l+1}(u), \]

\[ Y(v)=P_l^\lambda(u). \]

2) \(R=\infty\) (proton \(b\) has receded to infinity). The elliptic coordinates turn into parabolic ones:

\[ x=\xi-1\to \frac{r_a-z_a}{R}=\frac{x}{R},\quad \eta+1\to \frac{r_a-z_a}{R}=\frac{y}{R}, \]

\[ \varphi=\varphi,\quad \varepsilon=\sqrt{-2E}. \]

The equation becomes the problem of the hydrogen atom in parabolic coordinates:

\[ \frac{d}{dx}\left(x\frac{dX}{dx}\right)+\left(\frac{1}{2}Ex+1-\frac{\lambda^2}{4x}\right)X=0; \tag{1.4}\_\infty \]

\[ \frac{d}{dy}\left(y\frac{dY}{dy}\right)+\left(\frac{1}{2}E-\frac{\lambda^2}{4y}\right)Y=0, \tag{1.5}\_\infty \]

The solutions of these equations will be (for negative values of \(E\)):

\[ X(x)=e^{-\frac{\varepsilon x}{2}}x^{\frac{\lambda}{2}}L_{n_1+\lambda}^{\lambda}(\varepsilon x), \]

\[ n=n_1+n_2+\lambda+1; \]

\[ Y(\eta)=e^{-\frac{y}{2}}y^{\frac{\lambda}{2}}L_{n_2+\lambda}^{\lambda}(y), \]

\[ \left. \begin{array}{l} n_1\\ n_2 \end{array} \right\} \quad \text{parabolic quantum numbers.} \]

We have obtained solutions for two values of the parameter \(R=0\) and \(R=\infty\). In order to obtain an idea of the character of the solutions for finite \(R\), let us make use of the concept of nodal surfaces of the \(\psi\)-function. A nodal surface of the \(\psi\)-function is a surface at whose points the \(\psi\)-function vanishes. Consider the case \(R=0\). Denote by \(u_i, v_i\) the roots of the equations

\[ X(u_i)=0, \]

\[ Y(v_i)=0. \]

These roots give us the spherical and conical nodes of the function \(\varphi\) for the value \(R=0\). Let us count the number of spherical nodes:

\[ X(u)=e^{-u}u^l L_{n+l}^{2l+1}(u). \]

The value \(u=\infty\) is a node lying at infinity,

\[ \text{''}\quad u=0 \text{ is an } l\text{-fold node.} \]

In addition, \(L_{n+l}^{2l+1}(u)\) has \(n_r\) nodes.
Altogether there are \(n_r+l+1=n\) spherical nodes, of which one is at infinity and \(l\) have coalesced into a point.

The principal quantum number \(n\) is the number of all spherical nodes. Let us now count the number of conical nodes:

\[ P_l^\lambda(\cos\vartheta)=\sin^\lambda\vartheta\, \frac{d^\lambda P_l(\cos\vartheta)}{(d\cos\vartheta)^\lambda}, \]

where \(P_l\) is a polynomial of degree \(l\).

We have \(l-\lambda\) nodes that are the roots of the polynomial

\[ P_l^{(\lambda)}(\cos\vartheta)=0. \]

\[ \text{''}\quad \lambda \text{ nodes of the function }(\sin\vartheta)^\lambda, \]

which have coalesced into the \(z\)-axis.

Altogether there are \(l\) conical nodes, of which \(\lambda\) have coalesced into the straight line through \(o\) and \(z\).

Let us now imagine that we move the nuclei apart along the \(z\)-axis. The nodes will be deformed: the spherical ones turn into ellipsoidal ones, and the conical ones into hyperboloidal ones. The spherical nodes that have coalesced into a point will turn into ellipsoids degenerated into the segment between the foci. The degenerate conical nodes into straight lines going from

foci to infinity. To characterize the functions \(X(\xi)\) and \(Y(\eta)\) we may retain the quantum numbers \(n, l, \lambda\), writing

\[ \psi(\xi,\eta,\varphi)=X_{nl}(\xi)Y_l^\lambda(\eta)e^{i\lambda\varphi}, \]

where the indices \(n, l\) denote the number of nodes. This notation is convenient, since it makes it possible to see into which state an electron moving in the field of two centers passes when the nuclei are brought together.

Let us now follow the deformation of the nodes as the distance \(R\) is increased to infinity. We shall imagine that one of the centers recedes while the other remains fixed.

The ellipsoidal nodes will turn into paraboloids intersecting the \(z\)-axis outside the molecule. The hyperboloidal nodes into paraboloids,

Fig. 1.

intersecting the \(z\)-axis between the centers. Our problem again turns into a one-center problem, and the state of the electron can again be described by a hydrogen function, this time in parabolic coordinates.

Sometimes it is convenient, instead of the numbers \(n, l\), to introduce the numbers \(n_\xi\) and \(n_\eta\), denoting the number of nondegenerate nodes (ellipsoidal and hyperboloidal). Obviously, when the nuclei are brought together,

\[ n_\xi \to n_r = n-l-1, \]

\[ n_\eta \to l-\lambda. \]

When nucleus \(b\) is removed to infinity, the number \(n_\xi\) turns into the number \(n_1\) of nondegenerate parabolic nodes \(\xi=\xi_i\). As for the number \(n_\eta\), it becomes \(2n_2\) if the function \(Y(\eta)\) is even, and \(2n_2+1\) if \(Y(\eta)\) is odd. We obtain as the solution:

\[ X(x)=e^{-\frac{x}{2}}x^{\frac{\lambda}{2}}L_{n_1+\lambda}^{\lambda}(x), \]

\[ Y(y)=e^{-\frac{y}{2}}y^{\frac{\lambda}{2}}L_{n_2+\lambda}^{\lambda}(y). \]

To survey the connection between the quantum numbers, let us write out once more their mutual relation:

\[ \boxed{ \begin{gathered} n-l-1 \to n_{\xi}\to n_{1},\\[3pt] l-\lambda \to n_{\eta} \begin{cases} \to 2n_{2}\quad (\text{even } n_{\eta}),\\ \to 2n_{2}+1\quad (\text{odd } n_{\eta}), \end{cases}\\[3pt] R=0,\qquad R\ne 0,\qquad R=\infty . \end{gathered} } \]

Now we introduce the new important concept of the evenness and oddness of a state.

We say that the function \(\psi(x,y,z)\) is even if, upon replacing \(x,y,z\) by \(-x,-y,-z\), the function remains unchanged, and odd if it changes sign. We shall denote this by the symbols \(g\) and \(u\). Thus:

\[ \psi_g(x,y,z)=\psi_g(-x,-y,-z), \]

\[ \psi_u(x,y,z)=-\psi_u(-x,-y,-z). \]

It is easy to see that every hydrogen function with even \(l\) is even, and with odd \(l\) is odd. Indeed, for hydrogen we have:

\[ F(r)\,r^l P_l^\lambda(\cos\vartheta)e^{i\lambda\varphi} = F(r)\,V_l(x,y,z), \]

where \(V(x,y,z)\) is a homogeneous polynomial of degree \(l\) in \(x,y,z\), satisfying the equation \(\Delta V_l=0\). Passing to the two-center problem, we note that, by placing the origin of coordinates at the center of the molecule, under the substitution

\[ x\to -x,\quad y\to -y,\quad z\to -z, \]

we obtain

\[ \xi\to \xi,\quad \eta\to \eta,\quad \varphi\to \varphi+\pi, \]

which is equivalent to the substitution:

\[ Y_l^\lambda(\eta)e^{i\lambda\varphi} = Y_l^\lambda(-\eta)e^{i\lambda(\varphi+\pi)} = (-1)^\lambda Y_l^\lambda(-\eta)e^{i\varphi} \]

for an even function, or

\[ Y_l^\lambda(\eta)e^{i\lambda\varphi} = -\,Y_l^\lambda(-\eta)e^{i\lambda(\varphi+\pi)} = (-1)^{\lambda+1}Y_l^\lambda(-\eta)e^{i\varphi} \]

for an odd function.

Since \(Y_l^\lambda(\eta)\) has \(l-\lambda\) nodes between the protons, it is again obvious that for even \(l\) we have \(\psi_g\), and for odd \(l\), \(\psi_u\). Let us determine when the \(\psi\)-function will have a nodal plane perpendicular to the \(z\)-axis and passing through the center. We have:

\[ Y_l(0)=(-1)^\lambda Y_l^\lambda(0)\quad (\text{for an even } \psi\text{-function}), \]

\[ Y_l(0)=(-1)^{\lambda+1}Y_l^\lambda(0)\quad \text{for an odd } \psi\text{-function}. \]

Consequently,

\[ Y_l^\lambda(0)=0 \begin{cases} \text{for odd } \lambda \text{ for even } l,\\ \text{for even } \lambda \text{ for odd } l. \end{cases} \]

With the nodal plane, therefore, the electron states will be \(\sigma_u, \pi_g, \delta_u\). Without a nodal plane, \(\sigma_g, \pi_u, \delta_g\). This result is important for us to note.

For denoting the terms of an electron in the field of two centers one may use the numbers \(n, l, \lambda\). However, another notation has become established in spectroscopy. Instead of the numbers \(l=0, 1, 2, 3\ldots\) in atomic theory one writes the letters \(s\)-, \(p\)-, \(d\)-states. Therefore, instead of

\[ \left( \begin{array}{c} n,\ l,\ \lambda\\ 2,\ 1,\ 1 \end{array} \right) \]

one writes \(2p\pi\). There is no longer any need to indicate the parity, since it is evident from the parity of \(l\).

Exact solution for \(\mathrm{H}_2^+\)

Of the two equations (1.4) and (1.5), the second equation is solved comparatively easily. Its solution does not change its character if we let \(R\) approach zero, becoming in the limit the spherical function \(P_l^\lambda(\eta)\). Therefore it is natural to expand \(Y_l^\lambda(\eta)\) in a series in these functions:

\[ Y_l^\lambda(\eta)=\sum_{\sigma=\lambda}^{\sigma=\infty} a_\sigma(R)\,P_\sigma^\lambda(\eta), \tag{1.6} \]

where the expansion coefficients depend on \(R\). For small values of \(R\), evidently, all coefficients \(a_\sigma(R)\), except for one \(a_l(R)\), are small:

\[ R\to 0,\quad a_\sigma(R)=0\ \text{for}\ \sigma\ne l;\quad a_l(R)=1. \]

Substituting the expansion (1.6) into the equation and using the properties of the functions \(P_\sigma^\lambda(\eta)\), we obtain a system of equations for \(a_\sigma(R)\) and \(A(R)\). These systems can be solved by successive approximations, taking in the first approximation all \(a_\sigma(R)=0\), except \(a_l(R)\). In practice it is sufficient to determine all \(a_\sigma(R)\) in the second approximation.

As for the equation for \(X_{nl}(\xi)\), its solution changes its character depending on whether \(R\) is exactly equal to zero or differs from it. The boundary conditions require finiteness for \(1<\xi<\infty\), including the boundaries. The point \(\xi=1\) is a singular point of the equation. Therefore let us pass to the function \(f(\xi)\), regular at the point \(\xi=1\):

\[ X(\xi)=(\xi^2-1)^{\frac{\lambda}{2}} f(\xi). \]

Instead of the variable \(\xi\) it is convenient to pass to the parabolic coordinate \(x\), related to \(\xi\) by the condition:

\[ \xi-1=\frac{x}{R\varepsilon}, \qquad \text{where } \varepsilon=\sqrt{-2E}. \]

We obtain:

\[ X\left(1+\frac{x}{R\varepsilon}\right) = \left(\frac{2x}{R\varepsilon}+\frac{x^2}{R^2\varepsilon^2}\right)^{\frac{\lambda}{2}} f\left(1+\frac{x}{R\varepsilon}\right). \]

or changing the normalization:

\[ X\left(1+\frac{x}{R_e}\right) = \left(x+\frac{x^2}{2R_e}\right)^{\frac{\lambda}{2}} f\left(1+\frac{x}{R_e}\right). \]

As \(R \to \infty\) our equation passes into equation \((1.4)_\infty\) with the solution

\[ \lim_{R\to\infty} \left(x+\frac{x^2}{2R_e}\right)^{\frac{\lambda}{2}} F\left(1+\frac{x}{R_e}\right) = x^{\frac{\lambda}{2}} e^{-\frac{x}{2}} L_{n_1+\lambda}^{\lambda}(x). \]

Therefore we may expand \(f\left(1+\frac{x}{R_e}\right)\) in a series in the functions \(e^{-\frac{x}{2}}L_{\sigma+\lambda}^{\lambda}(x)\):

\[ f\left(1+\frac{x}{R_e}\right) = \sum_{\sigma=0}^{\sigma=\infty} b_\sigma(R)\, e^{-\frac{\varepsilon x}{2}} L_{\sigma+\lambda}^{\lambda}(\varepsilon). \tag{1.7} \]

For large values of \(R\), evidently, all coefficients except one \(b_{n_r}(R)\) are small:

\[ b_\sigma(R)=0,\ \sigma \ne n_r,\quad b_{n_r}(R)=1,\quad R\to\infty. \]

Substituting the expansion (1.7) into equation (1.4) and the value \(A(R)\) from (1.6), and using the properties of the functions \(L_{\sigma+\lambda}^{\lambda}(x)\), we obtain for \(b_\sigma(R)\) a system of equations which can be solved approximately. In the first approximation all \(b_\sigma(R)=0\) are set, with the exception of \(b_{n_r}(R)\). In practice it is sufficient to solve the problem in the second approximation. In view of the fact that we expand the desired functions in a series in the orthogonal functions \(P_j^\lambda(\eta)\), \(L_{\sigma+\lambda}^{\lambda}(x)\), regular throughout the entire range of variation \(-1<\eta<1,\ 0<x<\infty\), we may be certain that the desired functions will satisfy the boundary conditions.

The reader interested in the practical computation is referred to Hylleraas’s original work.

Approximate solution for \(H_2^+\)

We have shown how to arrive at an exact solution of the \(H_2^+\) problem. We now turn to approximate solutions and consider a method similar to that applied by Heitler and London to the hydrogen molecule.

The Schrödinger equation will be:

\[ \left\{-\frac{1}{2}\Delta-\frac{1}{r_a}-\frac{1}{r_b}\right\}\psi=E\psi . \tag{1.8} \]

Suppose that the electron is located near nucleus \(a\). Then the action on it of nucleus \(b\) may be neglected and, as an approxim-

of the solution, to write the hydrogen function \(u(q)\), satisfying the equation:

\[ \left\{-\frac{1}{2}\Delta-\frac{1}{r_a}\right\}u(q)=E_0u(q). \tag{1.9} \]

But the electron may also be located at nucleus \(b\). An approximate solution for (1.8) will be the function \(v(q)\), satisfying the equation:

\[ \left\{-\frac{1}{2}\Delta-\frac{1}{r_b}\right\}v(q)=E_0v(q). \tag{1.10} \]

If both functions \(u\) and \(v\) belong to one and the same energy value \(E_0\) of the hydrogen atom, then the function

\[ a u(q)+b v(q) \tag{1.11} \]

will be an approximate solution of equation (1.8).

Equation (1.8) is symmetric with respect to \(r_a\) and \(r_b\). Therefore the function (1.11) must be symmetric or antisymmetric in \(r_a\) and \(r_b\). We shall obtain two zero-approximation functions:

\[ w_s=c u(q)+v(q), \]

\[ w_a=c[u(q)-v(q)] \]

for our problem. In order to obtain the approximate energy, we must calculate the integrals:

\[ E_{s,a}= \frac{\int \{u(q)\pm v(q)\}\,\mathrm{H}\,\{u(q)\pm v(q)\}\,d\tau} {\int \{u(q)\pm v(q)\}\{u(q)\pm v(q)\}\,d\tau} = E_0+\frac{C\pm R}{1\pm S}, \]

where the following notation has been introduced:

\[ C=\int \frac{|u(q)|^2}{r_b}\,d\tau =\int \frac{|v(q)|^2}{r_a}\,d\tau, \]

\[ R=\int \frac{u(q)v(q)}{r_a}\,d\tau =\int \frac{u(q)v(q)}{r_b}\,d\tau, \]

\[ S=\int uv\,d\tau. \]

The integral \(S\) characterizes the degree of nonorthogonality of the functions \(u\) and \(v\). As \(R\to\infty\) it naturally tends to zero.

The integral \(C\) is the Coulomb interaction of the “foreign” nucleus with the electron.

As for the integral \(R\), it admits no classical interpretation and is analogous to the exchange integral of the Heitler and London theory. Here we have no exchange, since the question concerns only one electron. In the expression for the energy this integral causes the splitting of the degenerate term into two terms, even and odd.

It is obvious that the method set forth is applicable only for large distances between the nuclei. This follows at least from the fact that for small \(R\) the integral \(S\) tends to unity (\(u\to v\)) and the splitting of the term increases without bound.

When the distance is increased, the state of the electron tends toward the state of a separate atom. If we have taken as the initial function the hydrogen function with quantum numbers \(n, l\), and \(\lambda\), then the proper functions

\[ u(q)+v(q) \]

will approach the term \((n,l,\lambda)\) without a node,

\[ u(q)-v(q) \]

will approach the term \((n,l,\lambda)\) with a node.

The integrals \(S, C\), and \(R\) have been calculated.

The \(S\)-integrals are always positive, while the \(C\)- and \(R\)-integrals are negative.

The case we have analyzed is of fundamental importance for questions of the chemical bond, which in the present case is realized by a single electron, whereas for the realization of a chemical bond according to the spin-valence theory the presence of at least two electrons interacting by their spins is required.

This circumstance enables the molecular theory of the chemical bond to adopt a new point of view, differing from the point of view of spin valence and including the latter as a special case.

We see that an electron in the field of two nuclei may be in two different states: “with a node” or “without a node,” which possess different energies. Some states are characterized by the fact that the \(\psi\)-function vanishes at all points of the plane perpendicular to the axis of the molecule and passing through its center. These, as we have shown, are the \(\sigma_u\)-, \(\pi_g\)-, \(\delta_u\)-states. To these states there corresponds the minus sign in the formula for the energy of \(H_2^+\). They give repulsion. Conversely, those states which have no node, namely \(\delta_g\), \(\pi_u\), \(\delta_g\), give attraction. For this reason Herzberg calls the former antibonding states and the latter bonding states. In the theory set forth here, what is responsible for the chemical bond is the absence or presence of a node between the nuclei.

Fig. 2.

2. The \(H_2\) molecule.

Before passing to the molecular theory of the hydrogen molecule, let us recall the method of Heitler and London, with which we shall have to compare the molecular theory. The Schrödinger equation will be:

\[ H\psi=\left\{-\frac{1}{2}\Delta_1-\frac{1}{2}\Delta_2-\frac{1}{r_{a1}}-\frac{1}{r_{a2}}-\frac{1}{r_{b1}}-\frac{1}{r_{b2}}+\frac{1}{r_{12}}\right\}\psi=E\psi. \tag{2.1} \]

We have, as before, included the proton interaction \(\dfrac{1}{R}\) in the total energy \(E(R)\).

As an approximate solution one may write the function

\[ u(1)v(2), \tag{2.2} \]

expressing that electron 1 is at nucleus \(a\), and electron 2 at nucleus \(b\). This solution belongs to the eigenvalue \(E_0\), equal to the sum of the energies of the two atoms. In addition to this solution, the obvious solution belonging to the same energy \(E_0\) will be the function

\[ u(2)v(1), \tag{2.3} \]

expressing that electron 1 is at nucleus \(b\), and electron 2 at nucleus \(a\).

Thus any linear combination of the two solutions will also be a solution of our problem:

\[ \mathfrak{w}=a u(1)v(2)+b u(2)v(1). \tag{2.4} \]

By virtue of the symmetry of the problem, the function \(\mathfrak{w}\) must be either symmetric or antisymmetric in the electrons, and in the zero approximation we obtain:

\[ \begin{aligned} \mathfrak{w}_{+}&=u(1)v(2)+u(2)v(1),\\ \mathfrak{w}_{-}&=u(1)v(2)-u(2)v(1). \end{aligned} \tag{2.5} \]

In order to obtain the energy in this approximation, we must calculate the integral:

\[ E_{+,-}= \frac{ \int \{u(1)v(2)\pm u(2)v(1)\}\,H\,\{u(1)v(2)\pm u(2)v(1)\}\,d\tau_1\,d\tau_2 }{ \int |\{u(1)v(2)\pm u(2)v(1)\}|^2\,d\tau_1\,d\tau_2 } = \]

\[ =E_0+\frac{C\pm A}{1\pm S^2}, \tag{2.6} \]

where the following notation has been introduced:

\[ S=\int u(1)v(1)\,d\tau \]

as in the problem of \(\mathrm{H}_2^+\),

\[ C=\int \frac{|u(1)|^2|v(2)|^2\,d\tau_1\,d\tau_2}{r_{12}} -\int \frac{|u(1)|^2|v(2)|^2}{r_{a_2}} -\int \frac{|u(1)|^2|v(2)|^2}{r_{b_1}}, \]

(the Coulomb interaction of the electron clouds with each other and with the nuclei),

\[ A=\int u(1)v(2)u(2)v(1)\left\{\frac{1}{r_{12}}-\frac{1}{r_{a_2}}-\frac{1}{r_c}\right\}\,d\tau_1\,d\tau_2. \]

The last integral, the so-called exchange integral, has, as is known, no classical analogue. It corresponds to a certain probability of the electrons exchanging places. Calculation shows that the integrals \(C\) and \(A\) are negative.

Therefore the term corresponding to the even function will lie below the term corresponding to the odd one. Whereas Heitler–London see the cause of the chemical bond in the antisymmetry of the spin corresponding, by the Pauli principle, to the even function, molecular theory, in accordance with the problem for \(\mathrm{H}_2^+\), sees the cause of the chemical bond in the fact that in the case of the even function there is no node between the nuclei, while in the case of the odd function there is such a node.

Let us now turn to the molecular theory for \(\mathrm{H}_2\).

We may regard the problem of \(\mathrm{H}_2\) as analogous to the problem of helium in atomic theory. We have two molecular electrons moving in the field of two protons. The interaction of the two electrons with one another shall be treated as a perturbation. Suppose that the spins of both electrons are different. Then, by the Pauli principle, they may be in one molecular orbit. Let this orbit be \(\mathfrak g\). For the first electron we have the molecular function

\[ w(1)=u(1)+v(1), \]

and for the second the same:

\[ w(2)=u(2)+v(2), \]

for the molecule we shall have the symmetric function:

\[ w(1)w(2)=(u(1)+v(1))(u(2)+v(2))= \]

\[ =\{u(1)v(2)+v(1)u(2)\}+\{u(1)u(2)+v(1)v(2)\}. \]

The first part of the molecular function is precisely the Heitler–London function. As for the second term, it describes a state of the molecule which, when the protons are separated, passes into an ionic state \((\mathrm{H}^+—\mathrm{H}^-)\), in which both electrons are at one nucleus. The function \(w(1)w(2)\) is, therefore, a superposition of two states: homeopolar and heteropolar. At infinite \(R\) the energy of the ionic state lies much higher than the energy of the homeopolar state. The difference between them is equal to:

\[ (\text{ionization energy of } \mathrm{H})-(\text{electron-affinity energy of } \mathrm{H}) \]

\[ (1.0-0.18)\,Ry. \]

When the nuclei approach, however, we gain much energy corresponding to the Coulomb interaction of the ions. At equilibrium the ionic term lies only \(0.2\,Ry\) above the heteropolar one. In view of this, the hydrogen molecule is a typical homeopolar molecule, and therefore the Heitler–London method is justified for it. In the case where both terms are close to one another, the molecular method corresponds more closely to reality. The best description of the hydrogen molecule will be obtained if we consider the linear superposition of both states:

\[ w=\sqrt{1-\alpha^2}\,(u(1)v(2)+v(1)u(2))+\alpha(u(1)u(2)+v(1)v(2)) \]

and define \(a\) so that the integral

\[ \frac{\int w \mathrm{H} w \, d\tau_1\, d\tau_2} {\int w w \, d\tau_1\, d\tau_2} \]

gives a minimum. In this case \(a^2\) is found to be equal to 0.37 and gives the probability of the ionic state of the molecule. From these considerations we see how closely intertwined are the homeopolar and heteropolar properties of the molecule.

Homeopolar and heteropolar terms

Fig. 3.

If both electrons are not on one orbital, but on two different ones, \(\sigma_g\) and \(\sigma_u\), then both spin arrangements are possible. For the triplet we have the following state of the molecule:

\[ \sigma_g(1)\sigma_u(2)-\sigma_g(2)\sigma_u(1), \]

for the singlet:

\[ \sigma_g(1)\sigma_u(2)+\sigma_g(2)\sigma_u(1), \]

or, writing out in detail,

\[ \begin{aligned} \text{triplet}\quad &\{u(1)+v(1)\}\{u(2)-v(2)\}\\ &-\{u(2)+v(2)\}\{u(1)-v(1)\} = \{v(1)v(2)-u(1)v(2)\}, \end{aligned} \]

\[ \begin{aligned} \text{singlet}\quad &\{u(1)+v(1)\}\{u(2)-v(2)\}\\ &+\{u(2)+v(2)\}\{u(1)-v(1)\} = u(1)u(2)-v(1)v(2). \end{aligned} \]

The first of these corresponds to the repulsive term of the Heitler–London theory and gives no potential minimum.

The second, singlet term corresponds to the so-called \(B\)-term of the \(\mathrm{H}_2\) molecule.

The detailed calculation, carried out by Hylleraas, reduces to the following results.

Configuration, term	\(R_0\)	Dissociation energy
\(1\sigma_g\,1\sigma_g\) singlet	1.35	4.37 experiment — \(X\)-term of Heitler–London
\(1\sigma_g\,1\sigma_g\) singlet	1.42	4.42 theory — \(X\)-term of Heitler–London
\(1\sigma_g\,2\sigma_g\) triplet	1.85	2.89 experiment — excited term \(H_3\)
\(1\sigma_g\,2\sigma_g\) triplet	2.00	2.90 theory — excited term \(H_3\)
\(1\sigma_g\,2\sigma_u\) singlet	2.3	3.19 experiment — \(B\)-term
\(1\sigma_g\,2\sigma_u\) singlet	2.5	3.37 theory — \(B\)-term
\(1\sigma_g\,2\sigma_u\) triplet	\(\infty\)	— \(^{3}\Sigma\)-term of Heitler–London

As we see, the experimental result agrees excellently with the theory.

3. Systematics of Molecular Terms

The energy state of an atom, in addition to the magnitude of the energy, is also characterized by the values of two vectors \(L\) and \(S\).

The first of these denotes the orbital moment of the atom, i.e. the vector sum of the orbital moments \(l_i\) of all the electrons entering into it. It can have only integral values \(L=0, 1, 2,\ldots\). Depending on the values of \(L\), the states of an atom are denoted in spectroscopy as \(S\)-, \(P\)-, \(D\)-states. The second vector denotes the vector sum of all the spins of the individual electrons. It can have integral values \(S=0, 1, 2, 3,\ldots\) (for an even number of electrons) and half-integral values \(\frac{1}{2}, \frac{3}{2}, \frac{5}{2},\ldots\) (for an odd number of electrons). The number \(2S+1\) gives the multiplicity of the term. For \(S=0\) we have singlets, for \(S=\frac{1}{2}\) doublets, for \(S=1\) triplets, etc. The multiplicity is denoted by a superscript index. For example, a \(^{3}P\)-term means \(L=1\), \(S=1\). A \(^{4}S\)-term means \(L=0\), \(S=\frac{3}{2}\), etc.

Fig. 4.

As the atomic theory of chemical bonding shows, these two vectors are responsible for the chemical bond. The interaction of spin moments is the source of the forces of spin valence, and the interaction of orbital moments is the source of the forces of orbital valence.

When atoms combine into a molecule, the vectors \(L\) and \(S\) behave differently.

The spin moments of atoms \(A\) and \(B,\ldots\), denoted by \(S_a\) and \(S_b\), are combined according to the rules of quantum vector addition into the resultant spin moment of the molecule:

\[ S_{ab}=S_a+S_b,\; S_a+S_b-1,\ldots |S_a-S_b|. \]

Since we assume that the interaction between \(L\) and \(S\) is very small (of the order of the fine structure), the electric field of the nuclei has no effect on the spin moment of the molecule, and the vector \(S\) may have all directions in space, as in the case of an atom.

As for \(L\), the orbital moment of the molecule will no longer be a constant of the motion as in the atom, but will precess about the axis connecting the nuclei. On the basis of the general principles of quantum mechanics, only the components of the vector \(L\) along the molecular axis will have a constant value. These are customarily denoted by \(\Lambda\).

Let us see how many different molecular terms we obtain when atoms with vectors \(L_a\) and \(L_b\) are combined. For the vector \(L_{ab}\) we obtain \(|L_a>L_b|\):

\[ L_a+L_b,\; L_a+L_b-1,\ldots |L_a-L_b|; \]

in all, \(2L_b+1\) vectors.

Each of them is quantized in the direction of the axis connecting the nuclei. We obtain for \(\Lambda\) the following values:

\[ \Lambda = L_a+L_a,\; L_a+L_a-1,\ldots 0, \]
\[ L_a+L_a-1,\ldots 0, \]
\[ L_a-L_b,\; L_a-L_b-1,\ldots 0. \]

Molecular terms for which \(\Lambda\) is \(0, 1, 2, 3\), etc., by analogy with atomic terms, are denoted by \(\Sigma\)-, \(\Pi\)-, \(\Delta\)-, etc. terms.

Examples: An excited N atom is in the state \({}^{2}D\).
A normal O atom is in the state \({}^{3}P\).

We have:

\[ L_N=2,\quad S_N=\frac{1}{2}, \]

\[ L_O=1,\quad S_O=1. \]

Let us first determine the multiplicity of the terms of the NO molecule. We have:

\[ S_N+S_O,\; S_NS_O-1,\ldots \]

\[ \frac{3}{2},\qquad \frac{1}{2} \]

The terms will be quartets and doublets.

To determine the possible values of \(\Lambda\), we first find \(L_{\mathrm{NO}}\). We have:

\[ L_{\mathrm{NO}}=3,\; 2,\; 1. \]

These three vectors are quantized in the direction of the molecular axis, and we have:

\[ \Lambda = 3,\, 2,\, 1,\, 0 \]
\[ 2,\, 1,\, 0 \]
\[ 1,\, 0 \]

We have three \(\Sigma\)-terms, three \(\Pi\)-terms, two \(\Delta\)-terms and one \(\Phi\)-term. Each of them may be a doublet or a quartet.

Altogether we have 18 terms:

\[ {}^{4}\Sigma\;{}^{4}\Sigma\;{}^{4}\Sigma,\qquad {}^{4}\Pi\;{}^{4}\Pi\;{}^{4}\Pi,\qquad {}^{4}\Delta\;{}^{4}\Delta\;{}^{4}\Phi, \]

\[ {}^{2}\Sigma\;{}^{2}\Sigma\;{}^{2}\Sigma,\qquad {}^{2}\Pi\;{}^{2}\Pi\;{}^{2}\Pi,\qquad {}^{2}\Delta\;{}^{2}\Delta\;{}^{2}\Phi. \]

In diatomic molecules with identical nuclei one may further classify the terms according to the behavior of the wave function under reflection in the center of the molecule into even and odd terms, just as we did for the terms of individual electrons.

We shall have \(\Sigma_g\)-, \(\Sigma_u\)-, \(\Pi_g\)-, \(\Pi_u\)-, etc. terms.

In addition to this symmetry, for the molecule there is also significance in the behavior of the wave function under reflection in a plane passing through the molecular axis. We have an operation under which \(\varphi_i\) goes into \(-\varphi_i\), for each electron. This means that \(\lambda_i\) goes into \(-\lambda_i\), or \(\Lambda=\Sigma\lambda_i\) goes into \(-\Lambda\). For \(\Pi\)-, \(\Delta\)-, etc. terms this is immaterial, since they are all expressed with respect to the sign of \(\Lambda\). As for the \(\Sigma\)-term, under this operation \(\psi\) either changes sign (\(\Sigma^{-}\)-term) or remains unchanged (\(\Sigma^{+}\)-term). For an electron this does not occur, since by definition the \(\sigma\)-state does not depend on \(\varphi\), and therefore every \(\sigma\)-state is \(\sigma^{+}\).

We have constructed the molecule from ready-made atoms. Let us now show how to determine the terms of a molecule starting from the molecular electrons.

One \(\sigma\)-electron gives a \({}^{2}\Sigma\)-term.

Two \(\sigma\)-electrons give a \({}^{1}\Sigma^{+}\)- and a \({}^{3}\Sigma^{+}\)-term if they are located in different orbits (the spins may be the same or different), and a \({}^{1}\Sigma^{+}\)-term if they are located in one orbit.

\[ \left. \begin{array}{l} \text{One }\sigma\text{-electron}\\ \text{One }\pi\text{-electron} \end{array} \right\} \quad \text{give }{}^{1}\Pi\text{ or }{}^{3}\Pi\text{ depending on the spins} \]

One \(\pi\)-electron gives a \({}^{2}\Pi\)-term.

Two \(\pi\)-electrons (in one shell). Let us write the wave functions for two electrons (in cylindrical coordinates):

\[ f(z_1,r_1)f(z_2,r_2) \left\{ \begin{array}{ll} e^{i\varphi_1},\, e^{i\varphi_2}, & \Lambda=+2,\\ e^{i\varphi_1},\, e^{-i\varphi_2}, & \Lambda=0,\\ e^{-i\varphi_1},\, e^{i\varphi_2}, & \Lambda=0,\\ e^{-i\varphi_1},\, e^{-i\varphi_2}, & \Lambda=-2. \end{array} \right. \]

The Pauli principle requires that the functions be either symmetric in the coordinates (and antisymmetric in the spins), or antisymmetric (and symmetric in the spins).

Therefore we shall have:

\[ (z_1 r_1) f(z_2 r_2) \begin{cases} e^{i\varphi_1},\ e^{i\varphi_2},\ \Lambda=2,\ S=0,\\ e^{-i\varphi_1},\ e^{-i\varphi_2},\ \Lambda=-2,\ S=0, \end{cases} \quad {}^1\Delta\text{-term}; \]

\[ (z_1 r_1) f(z_2 r_2) \begin{cases} e^{i\varphi_1},\ e^{-i\varphi_2}+e^{-i\varphi_1},\ e^{i\varphi_2},\ \Lambda=0,\ S=0\quad {}^1\Sigma^{+}\text{-term};\\ e^{i\varphi_1},\ e^{-i\varphi_2}-e^{-i\varphi_1},\ e^{i\varphi_2},\ \Lambda=0,\ S=1\quad {}^3\Sigma^{-}\text{-term}. \end{cases} \]

The \({}^1\Sigma^{+}\)-term remains unchanged under \(\varphi \to -\varphi\), while the \({}^3\Sigma^{-}\)-term changes sign. In the absence of interaction between the two electrons, all these terms are degenerate. When the interaction is included, they split, and, just as in atoms, the term of higher multiplicity, \({}^3\Sigma^{-}\), proves to be the lowest.

Three \(\pi\)-electrons (in one shell) give the same terms as one \(\pi\)-electron, which is lacking for complete filling of the shell (Lückenprinzip).

In constructing the terms of a molecule, the filled shells (a \(\sigma\)-shell of two electrons and a \(\pi\)-shell of four electrons, a \(\delta\)-shell of four electrons) may, like the filled shells of atoms, be disregarded.

4. General Theory of Diatomic Molecules

We now turn to the construction of the periodic system of diatomic molecules. Before proceeding to this problem, we must generalize our model and consider the motion of an electron in the field of two non-Coulomb centers. The Schrödinger equation will be:

\[ \left\{-\frac{1}{2}\Delta - V(r_a) - V(r_b)\right\}\psi = E\psi . \tag{4.1} \]

Just as in the theory of the atom we pass from the hydrogen atom to hydrogen-like atoms of the alkali metals, in which one outer electron moves in the non-Coulomb central field of other electrons forming a spherical shell, so in the theory of diatomic molecules we must pass to the consideration of the “hydrogen-like ion-molecule” molecules described by equation (4.1). In this way we shall, to a certain degree, approach the conditions prevailing in the molecule, taking into account the screening of the nuclear charges by the inner electrons. Thus, we shall assume that a certain part of the electrons forming the shell of the molecule remains in the molecule near its nuclei. If we recall the picture of the atomic theory of valence, we shall see that only the so-called outer valence electrons form a common shell and are situated on a molecular orbit. For example, considering the molecule \(\mathrm{N}_2\), it is natural to suppose that, of the 14 electrons, only 6 electrons (three from each of the original atoms) form the molecular shell—

shell, while the remaining 8 electrons are distributed equally between the nuclei, forming their \(K\)- and \(L_1\)-layers.

In molecular theory the matter is complicated by the fact that the distribution of electrons into atomic and molecular states depends on the distance. As the nuclei approach one another, the atomic shells are destroyed and the atomic electrons pass into molecular orbitals.

Equation (3.1) is not separable in elliptic coordinates. Therefore we cannot introduce elliptic quantum numbers, and we can only describe the term in two limiting cases: \(R=0\) and \(R=\infty\). In the first case we obtain a hydrogen-like atom with a certain value of the angular momentum \(l\). When the nuclei are pulled apart this angular momentum is quantized along the axis of the molecule with component values

\[ \Lambda = 0, 1, 2, \ldots \]

A simple calculation shows that the energies of the molecular states are arranged in the same order. Let us pass to the case \(R=\infty\). Let us remove nucleus \(b\) to infinity. We again obtain a hydrogen-like atom; moreover, for large values of \(R\) the electron may be either at nucleus \(a\) or at nucleus \(b\), as a result of which there is a splitting of the state into “without a node” and “with a node.” Let us arrange the states in order of increasing energy for the cases \(R=0\) and \(R=\infty\). The question arises as to which terms pass into one another when \(R\) changes. Here the law of conservation of nodes, which we analyzed in detail for the case of the molecule \(\mathrm{H}_2^+\), comes to our aid. Even terms can pass only into even ones. We obtain the following scheme. Let us note that the mutual spacing of the terms has deliberately not been reflected in our scheme. It resembles an electrical-wiring diagram, which is drawn when it is necessary to indicate which points of an apparatus are connected by a wire and their mutual distance is of no significance. Nevertheless, even this rough scheme will enable us to discern all the known regularities in molecules from \(\mathrm{H}_2\) to \(\mathrm{Ne}_2\) . . .

Correspondence of terms for \(R=0\) and \(R=\infty\)

Fig. 5.

Let us proceed to the construction of the system of molecules. Each line of the scheme symbolizes a molecular orbital. No more than two electrons can be placed on \(\sigma\)-orbitals (Pauli principle). On \(\pi\)-, \(\delta\)-, etc., orbitals, which are doubly degenerate \((+\lambda)\), there can be accom-

be accommodated by four electrons. We shall have shells consisting of two and four electrons.

The 1st electron is placed in the \(\sigma_g\)-orbital; it will be bonding.
Molecule \(H_2^+\).

The 2nd electron is placed in the \(\sigma_g\)-orbital; it will be bonding. Molecule \(H_2\). The bond energy of \(H_2\) is greater than the energy of \(H_2^+\), as is to be expected.

The 3rd electron is placed in the \(\sigma_u\)-orbital; it will be antibonding. Molecule \(He_2^+\). Two bonding electrons and one antibonding electron nevertheless give a certain bond. The molecule \(He_2^+\) was observed spectroscopically by Weizel.

The 4th electron is placed in the \(\sigma_u\)-orbital. We obtain the “molecule” helium \(He_2\). Since we have two bonding and two antibonding electrons, the chemical bond is absent.

Continuing to fill the levels of our molecule, we obtain \(Li_2\) with the electron configuration \(\sigma_g^2 \sigma_u^2 \sigma_g^2\). We have four bonding and two antibonding electrons. For large distances between the nuclei it is natural to assume that only the outer \(\sigma_g\)-electrons are molecular electrons, while the inner ones remain with their nuclei. We have the following notation for the configurations of \(Li_2\):

\[ \frac{R\text{-small}}{\sigma_g^2 \sigma_u^2 \sigma_g^2}, \qquad \frac{R\text{-large}}{(1s)^2(1s)^2 \sigma_g^2}. \]

In the latter case \(Li_2\) is a homologue of \(H_2\).

\(Be_2\). The electron configuration:

\[ \frac{R\text{-small}}{\sigma_g^2 \sigma_u^2 \sigma_g^2 \sigma_u^2}, \qquad \frac{R\text{-large}}{(1s)^2(1s)^2 \sigma_g^2 \sigma_u^2}. \]

\(Be_2\) is a homologue of \(He_2\).

\(B_2\). The electron configuration:

\[ \frac{R\text{-small}}{\sigma_g^2 \sigma_u^2 \sigma_g^2 \sigma_u^2 \pi_u^2}, \qquad \frac{R\text{-large}}{(1s)^2(1s)^2(2s)^2 \pi_u^2}. \]

We have an excess of two bonding electrons. Molecule \(B—B\). Such a molecule has not been observed spectroscopically.

We can relatively easily excite the atom \(B\). In this case the configuration will be:

\[ B_2.\quad \frac{R\text{-small}}{\sigma_g^2 \sigma_u^2 \sigma_g^2 \pi_u^4}, \qquad \frac{R\text{-large}}{(1s)^2(1s)^2 \sigma_g^2 \pi_u^4}. \]

We have an excess of six bonding electrons. Molecule \(B \equiv B\).

It is possible that, owing to the interaction of the electrons, the states \(B—B\) and \(B \equiv B\) are almost degenerate, or even \(B \equiv B\) lies lower than \(B—B\). In this case we encounter the presence of a “hole” in the molecular shell, analogous to a “hole” in an atomic shell, which causes, for example, the insertion of the elements of the iron series and a disruption of the principal trend in the periodic system of the elements.

C₂. Electronic configuration:

$$ \frac{R\ \text{small}}{\sigma_g^2\sigma_u^2\sigma_g^2\sigma_u^2\pi_u^4}, \quad \frac{R\text{-large}}{(1s)^2(1s)^2(2s)^2\pi_u^4}. $$

An excess of four bonding electrons. Molecule C=C.

C₂. The combination of two excited C atoms gives:

$$ \frac{R\text{-small}}{\sigma_g^2\pi_u^2\sigma_g^1\pi_u^4\sigma_g^2}, \quad \frac{R\text{-large}}{(1s)^2(1s)^2\sigma_g^2\pi_u^4\sigma_g^2}. $$

An excess of eight bonding electrons. Molecule C≡C. What was said about B applies literally also to C. In any case, the states C≡C and C=C are energetically very close.

N₂. Electronic configuration:

$$ \frac{R\ \text{small}}{\sigma_g^2\sigma_u^2\sigma_g^2\pi_u^4\sigma_g^2}, \quad \frac{R\text{-large}}{(1s)^2(1s)^2(2s)^2\pi_u^4\sigma_g^2}. $$

An excess of six bonding electrons. Molecule N≡N.

O₂. Electronic configuration:

$$ \frac{R\text{-small}}{\sigma_g^2\sigma_u^2\sigma_g^2\sigma_u^2\pi_u^4\sigma_g^2\pi_g^2}, \quad \frac{R\text{-large}}{(1s)^2(1s)^2(2s)^2\pi_u^4\sigma_g^2\pi_g^2}. $$

An excess of four bonding electrons. Molecule O₂.

Characteristic of the O₂ molecule is that the last $\pi_g$ layer, in which there are four electrons, is not filled. Both $\pi$-electrons therefore have the possibility of being placed on different orbitals of the $\pi$-layer of the molecule.

We have the following states of the molecule (see pp. 51—52):

$$ \Lambda=2,\quad S=0 \quad {}^1\Delta\text{-term}, $$

$$ \Lambda=0,\quad S=0 \quad {}^1\Sigma\text{-term}, $$

$$ \Lambda=0,\quad S=1 \quad {}^3\Sigma\text{-term}. $$

The lowest term will be the ${}^3\Sigma$-term. This explains the paramagnetism of oxygen. It is curious to note that an unfilled layer occurs in the B—B molecule.

But it is apparently very unstable and has not been found up to the present. Similarly we obtain the electronic configurations for F₂ and Ne₂.

F₂. We have an excess of two bonding electrons. Molecule F₂.

Ne. Bonding and antibonding electrons balance one another. Chemical bonding is impossible.

In the molecules Na₂, P₂, S₂, Cl₂, the same thing is repeated as has been said up to now, and we shall not present the analysis.

Let us note that, according to the molecular theory, the noble gases are inert not in principle, as in the Heitler–London theory, but because the energetic balance approximately reduces to zero. The question of how strong the bond or the repulsion is cannot be decided without taking into account the energy conditions in the molecule. All the more surprising is it that this rough scheme already proves sufficient to explain a great variety of facts, whereas in the spin-valence theory a calculation must be carried out in each separate case. Moreover, the molecular theory makes it possible to judge how the chemical bond increases or decreases upon excitation of the molecule. This is its great advantage.

5. Molecules with different nuclei

We have generalized our method to the case of diatomic molecules with different nuclei. The equation of our problem will be:

\[ \left\{-\frac{1}{2}\Delta\psi - V(r_a)-U(r_b)\right\}\psi=E\psi, \tag{5.1} \]

where \(U\) and \(V\) are two different potential functions. The problem will no longer have symmetry with respect to the nuclei. We shall seek the solution in the form:

\[ \psi=au(q)+bv(q), \tag{5.2} \]

where \(u(q)\) and \(v(q)\) are solutions of the equations:

\[ \left\{-\frac{1}{2}\Delta-U\right\}u=E_0u, \tag{5.3} \]

\[ \left\{-\frac{1}{2}\Delta-V\right\}v=E_0v. \]

We thus assume that both states \(u\) and \(v\) of the separate atoms have, at least approximately, the same energy value \(E_0\).

Substituting (4.2) into (4.1), and using (4.3), we obtain

\[ aE_0u-bE_0v-aVu-bUv=aEu+bEv. \]

Multiplying both sides scalarly from the left first by \(u\), then by \(v\), we obtain:

\[ a\{(E_0-E)-V_{11}\}-bU_{12}=0 \qquad V_{11}=\int uVu\,d\tau,\quad U_{12}=\int uUv\,d\tau. \tag{5.4} \]

\[ aV_{21}-b\{(E_0-E)-U_{22}\}=0 \qquad V_{21}=\int vVu\,d\tau,\quad U_{22}=\int vUu\,d\tau. \]

Here we have made the simplifying assumption that both nuclei are at such a distance from one another that both functions are practically orthogonal, i.e. \(\int uvd\tau\to 0\).

From (4.4) we obtain for \(E_0-E\) the following two values:

\[ E_0-E_+=-\frac{V_{11}+U_{22}}{2} +\sqrt{(V_{11}-U_{22})^2+4U_{12}V_{21}}, \tag{5.5} \]

\[ E_0-E_-=-\frac{V_{11}+U_{22}}{2} -\sqrt{(V_{11}-U_{22})^2+4U_{12}V_{21}}. \]

We see that both roots are real if only \(U_{12}\) and \(V_{21}\) have the same sign. Knowing the roots, we can easily determine the ratio \(b/a\); we obtain:

\[ \left(\frac{b}{a}\right)_+ =\frac{E_0-E_+-V_{21}}{u_{12}}, \qquad \left(\frac{b}{a}\right)_- =\frac{E_0-E_- - V_{21}}{u_{12}}. \]

It is easy to see that these two numbers have different signs. Therefore we obtain the following result: the molecular state of the electron will be described by a function of the form:

\[ u+\lambda v \quad \text{or} \quad u-\mu v, \]

where \(\lambda\) and \(\mu\) are two positive numbers determined from the secular equation (4.4). It is easy to see that for \(U(r)=V(r)\) we obtain \(\lambda=\mu\), i.e., the case of identical nuclei.

The molecular states described by the functions \(u+\lambda v\) have no node between the nuclei; following Herzberg, we call them bonding. The states described by the functions \(u-\mu v\) have a node between the nuclei; we shall call them antibonding.

What has been set forth enables us to understand why molecules with different nuclei, but the same number of electrons, possess similar physical properties and analogous spectra.

For example, CO and \(N_2\) both contain 14 electrons. They possess similar physical properties (melting point, boiling point). They have an equal effective diameter for the passage of electrons, which indicates the same electronic structure.

Spectroscopically, a whole series of molecules with 13 electrons is known: CN, \(N_2^+\), BO, \(CO^+\). As should be expected, they all have analogous spectra.

Molecules with 12 electrons are \(C_2\) and BN. Both substances crystallize in the graphite lattice, which indicates a common electronic structure.

6. Polyatomic molecules

In passing to polyatomic molecules, it is necessary strictly to distinguish the case when the nuclei are situated on one straight line from the case when they are situated in space. The first case differs in principle only little from the case of diatomic molecules. The component of the angular momentum remains an integral of the motion, and the concept of \(\sigma\)-, \(\pi\)-, \(\delta\)-electrons retains its meaning. We shall discuss here several examples of polyatomic molecules in order to explain, from the standpoint of the theory, certain properties of molecules known from experience.

The simplest (unstable) polyatomic molecule is the model of three hydrogen atoms analyzed by London. On it we can study the important phenomenon of saturation of valence forces: two atoms joined into a molecule repel a third atom.

The equation for the molecular orbital will be:

\[ \left\{-\frac{1}{2}\Delta-\frac{1}{r_a}-\frac{1}{r_b}-\frac{1}{r_c}\right\}\psi=E\psi, \tag{6.1} \]

where \(r_a, r_b, r_c\) denote the distances of the electron from the three protons. As in the case of diatomic molecules, we shall approximate

molecular function by means of the functions of the three atoms \(\varphi_a,\varphi_b,\varphi_c\):

\[ \psi_1=\alpha\varphi_a+\beta\varphi_b+\gamma\varphi_c, \tag{6.2} \]

where the numbers \(\alpha,\beta,\gamma\) (more precisely, their ratios) are determined from the secular equation by the perturbation method. We shall have one function in which all the numbers \(\alpha,\beta,\gamma\) are positive. On the molecular orbit described by this function there will be placed two of the three electrons. The third electron can no longer be accommodated on the bonding orbit and will fall onto the antibonding one, described by a function in which at least one of the numbers \(\alpha,\beta,\gamma\) is negative:

\[ \psi_2=\alpha'\varphi_a+\beta'\varphi_b-\gamma'\varphi_c. \tag{6.3} \]

Which of the functions carries the negative coefficient depends on the configuration of the atoms, since the numbers \(\alpha,\beta,\gamma\) depend on the distances \(r_{ab}, r_{ac}, r_{bc}\).

We see that in our problem two bonding and one antibonding electron are involved. The question arises: which of the possible valence schemes

\[ \begin{array}{ccc} \begin{array}{c} c\\[-2pt] \circ\\[-2pt] /\\[-2pt] \circ\quad\circ\\[-2pt] a\quad b \end{array} & \begin{array}{c} c\\[-2pt] \circ\\[-2pt] \\[-2pt] \circ\!-\!\circ\\[-2pt] a\quad b \end{array} & \begin{array}{c} c\\[-2pt] \circ\\[-2pt] \backslash\\[-2pt] \circ\quad\circ\\[-2pt] a\quad b \end{array} \end{array} \]

corresponds to the state of our molecule when two electrons are on the orbit \(\psi_1\) and one on the orbit \(\psi_2\)?

Let us recall that both states of the system, \(\psi_1\) and \(\psi_2\), at infinite separation between the nuclei correspond to one and the same energy state—the energy of three hydrogen atoms. Therefore, at large separation we are entitled to take linear combinations of both states:

\[ \begin{aligned} \psi_s&=\lambda\psi_1+\mu\psi_2 =(\lambda\alpha+\mu\alpha')\varphi_a+(\lambda\beta+\mu\beta')\varphi_b+(\lambda\gamma-\mu\gamma')\varphi_c,\\ \psi_a&=\lambda\psi_1-\mu\psi_2 =(\lambda\alpha-\mu\alpha')\varphi_a+(\lambda\beta-\mu\beta')\varphi_b+(\lambda\gamma+\mu\gamma')\varphi_c. \end{aligned} \tag{6.4} \]

and choose \(\lambda\) and \(\mu\) so that, approximately,

\[ \begin{aligned} \lambda\psi_1+\mu\psi_2&\approx a\varphi_a+b\varphi_b,\\ \lambda\psi_1-\mu\psi_2&\approx c\varphi_c. \end{aligned} \tag{6.5} \]

We shall have a state corresponding to the valence scheme: \(a—b\ c—\). The electrons of atoms \(a\) and \(b\) establish a bond; the electron of atom \(c\) remains at its own nucleus. We see that the valence state is a superposition of two electron states, of which one is bonding and the other antibonding. The functions \((\psi_s,\psi_a)\), in co-

…describing in the aggregate a pure valence state, will be good approximations only in the case where both states \(\psi_1\) and \(\psi_2\) are not too far removed from one another.

In the molecular theory of polyatomic molecules we encounter the very same phenomenon of “jumping” valence as in the theory of spin valence. Here we are dealing directly with the energy functions \(\psi_1\) and \(\psi_2\), from which, by linear combination, the valence functions are constructed. In the theory of spin valence, on the contrary, we start from the valence functions and from them, by linear combination, construct the energy functions.

In a polyatomic molecule it is natural to speak of a localized bond between atoms \(a\) and \(b\) in the case where the molecular function can be represented in the form:

\[ \psi = a \varphi_a + b \varphi_c . \]

The extent to which such a function describes a molecular orbit depends on how the remaining atoms are situated. If they are all far from \(a\) and \(b\), then obviously we have an admissible approximation. When other atoms approach, the states of these atoms are admixed to the state \(\psi\), and that “valence hopping” takes place which is so characteristic of all quantum chemistry.

Among polyatomic molecules we shall touch especially on hydrides. Owing to the smallness of the H nuclei, one may in the first approximation regard H as included in the nucleus of the neighboring atom and consider the appearance of a new center as a perturbation:

\[ \begin{aligned} \mathrm{CH} &\text{ isoelectronic with } \mathrm{N},\\ \mathrm{CH}_2 &\text{ ” } \mathrm{O},\\ \mathrm{CH}_3 &\text{ ” } \mathrm{F},\\ \mathrm{CH}_4 &\text{ “ } \mathrm{Ne}, \end{aligned} \]

and therefore their electronic structure may be regarded as the perturbed electronic structure of the corresponding atoms. From this point of view the boron hydrides are of interest:

\[ \begin{aligned} \mathrm{BH} &\text{ isoelectronic with } \mathrm{C}.\\ \mathrm{B}_2\mathrm{H}_2 &\text{ ” } \mathrm{C}_2,\\ \mathrm{B}_2\mathrm{H}_6 &\text{ ” } \mathrm{O}_2,\\ \mathrm{C}_2\mathrm{H}_4 &\text{ ” } \mathrm{O}_2, \end{aligned} \]

The molecules \(\mathrm{B}_2\mathrm{H}_2\) are unstable, as is the molecule \(\mathrm{C}=\mathrm{C}\).

Therefore two molecules \(\mathrm{BH}_3\) attract one another, although according to valence theory they are saturated.

One should expect a similarity in the properties of \(\mathrm{B}_2\mathrm{H}_6\) and \(\mathrm{C}_2\mathrm{H}_4\). In particu-

of the phenomenon of cis- and trans-isomerism in derivatives of \(B_2H_6\), just as we have it in derivatives of \(C_2H_4\).

In his works on polyatomic molecules, Hund shows that, from the point of view of molecular theory, it is also possible to explain the phenomenon of directed valence (the triangular form of \(H_2O\), pyramidal \(NH_3\), etc.). The phenomenon of directed valence has been elaborated in detail in the theory of Pauling and Slater. Not being able here to enter into a consideration of this theory, we cannot touch here upon Hund’s views, since we would need to compare them with Pauling’s results.

We shall finish our survey by interpreting the phenomenon of cis- and trans-isomerism, which shows how fruitful the theory set forth here is.

7. Cis- and trans-isomerism

We shall show, following E. Hückel, that molecular theory is also capable of explaining the existence of cis- and trans-isomers. Let us consider a typical case:

\[ \begin{array}{ccccc} & \mathrm{Cl} & & \mathrm{H} & \\ & \backslash & & / & \\ & \mathrm{C}=\mathrm{C} & & & \\ & / & & \backslash & \\ & \mathrm{H} & & \mathrm{Cl} & \end{array} \qquad \text{and} \qquad \begin{array}{ccccc} & \mathrm{Cl} & & \mathrm{Cl} & \\ & \backslash & & / & \\ & \mathrm{C}=\mathrm{C} & & & \\ & / & & \backslash & \\ & \mathrm{H} & & \mathrm{H} & \end{array} \]

In order to convert one isomer into the other, it is necessary to expend a certain energy. Therefore we say that the radical

\[ \begin{array}{c} \backslash \\ \mathrm{C}= \end{array} \]

does not possess free rotation. In order to investigate this case, let us imagine that the ethylene molecule

\[ \begin{array}{ccccc} \mathrm{H} & & \mathrm{H} \\ & \backslash & / & \\ & \mathrm{C}=\mathrm{C} & \\ & / & \backslash & \\ \mathrm{H} & & \mathrm{H} \end{array} \]

arises from an isoelectronic molecule of the type \(O=O\) by the action on \(O=O\) of four positively charged centers (hydrogen nuclei) situated in the plane of the drawing. In \(O=O\) we have two molecular \(\pi\)-electrons in the outer shell of the molecule. Each of them is described by a function of the form \(\psi(r_1,z_1)e^{\pm i\varphi}\). The double sign at \(\varphi\) is placed because of the degeneracy of the \(\pi\)-state. Let us include the perturbation arising from the four hydrogen nuclei. This perturbation will destroy the free rotation of our system, and the degenerate \(\pi\)-term will split into two terms. We must introduce a linear combination of the functions \(\psi(r_1,z_1)e^{i\varphi}\) and \(\psi(r_1,z_1)e^{-i\varphi}\). Instead of rotation about the axis there must be symmetry with respect to reflection in the plane of the drawing, i.e. under replacement of \(\varphi\) by \(-\varphi\). The state which is symmetric under reflection in the plane,

denote the symmetric one by \(\pi_+\), the antisymmetric one by \(\pi_-\). Obviously, we obtain the following linear combinations:

\[ \pi_+ = \psi(r_1 z_1)\cos\varphi,\qquad \pi_- = \psi(r_1 z_1)\sin\varphi . \]

We have the following functions for two molecular electrons (in the zeroth approximation without interaction):

State and function	Energy
\(\pi_+(1)\pi_+(2)\),	\(2\varepsilon_+\),
\(\pi_-(1)\pi_-(2)\),	\(2\varepsilon_-\),
\(\pi_-(1)\pi_+(2)\),	\(\varepsilon_+ + \varepsilon_-\),
\(\pi_-(2)\pi_+(1)\).	\(\varepsilon_+ + \varepsilon_-\).

The last state is degenerate. From it we form two states, symmetric and antisymmetric in the electrons. In the zeroth approximation we obtain:

\[ \pi_-(1)\pi_+(2)+\pi_-(2)\pi_+(1), \]

\[ \pi_-(1)\pi_+(2)-\pi_-(2)\pi_+(1). \]

With these functions we shall solve our perturbation problem.

We have four functions for two electrons, not interacting with one another, chosen so that, when the interaction and the perturbing action of the four nuclei are included, they give the best approximation. When the interaction of the electrons is included, we have, in first approximation, the energy:

State	Energy
\(\pi_+(1)\pi_+(2)\),	\(2\varepsilon_+ + C_{++}\),
\(\pi_-(1)\pi_-(2)\),	\(2\varepsilon_- + C_{--}\),
\(\pi_-(1)\pi_+(2)+\pi_-(2)\pi_+(1)\),	\(\varepsilon_+ + \varepsilon_- + C_{+-} + A_{+-}\),
\(\pi_-(1)\pi_+(2)-\pi_-(2)\pi_+(1)\).	\(\varepsilon_+ + \varepsilon_- + C_{+-} - A_{+-}\),

where \(C\) denotes the Coulomb interaction, and \(A\) the exchange interaction, as in the usual Heitler–London problem. Since we do not have exact expressions for the functions \(\psi(r_1 z_1)e^{\pm i\varphi}\), we cannot compute all these integrals exactly. However, from general considerations one can estimate their relative magnitude. It turns out that

\[ \pi_-\pi_- \text{ lies lower than the } \pi_+\pi_+ \text{ state,} \]

\[ \pi_-(1)\pi_+(2)-\pi_-(2)\pi_+(1) \text{ lies lower than the } \pi_-(1)\pi_+(2)+\pi_-(2)\pi_+(1) \text{ state.} \]

In order to decide which of the terms \(\pi_-(1)\pi_-(2)\) or \(\pi_+(1)\pi_-(2)-\pi_+(2)\pi_-(1)\) lies lower, a physical consideration comes to the rescue. The first term is symmetric in the coordinates, hence antisymmetric in the spins. Conversely, the second is symmetric in the spins. If it were the lowest term, then ethylene wo-

similarly, \(O_2\) should be paramagnetic. Consequently, the very lowest term will be \(\pi_{-}(1)\pi_{-}(2)\).

Let us now rotate the plane of the hydrogen nuclei by \(90^\circ\). Since in doing so \(\sin \varphi\) changes into \(\cos \varphi\), and conversely, the term \(\pi_{-}(1)\pi_{-}(2)\) changes into the term \(\pi_{+}(1)\pi_{+}(2)\). But for this one must expend energy equal to the difference of the terms

\[ \pi_{+}(1)\pi_{+}(2)-\pi_{-}(1)\pi_{-}(2). \]

We see how a comparatively simple argument explains the fact, well known to chemists, of cis- and trans-isomerism.

References

1. Hund F., Ztschr. f. Phys., 73, 1, 1931.
2. Hertzberg G., Ztschr. f. Phys., 57, 601, 1929.
3. Mulliken R., Phys. Rev. 33, 730, 1929.
4. Teller E., Ztschr. f. Phys., 61, 458, 1930.
5. Hylleraas E., Ztschr. f. Phys., 71, 739, 1931.
6. Hund E., Ztschr. f. Phys., 63, 719, 1930.
7. Hückel E., Ztschr. f. Phys., 60, 423, 1930.

General theory of molecules.

\(H_2\) ion.

\(H_2\) double bond.

Review articles:

1. Herzberg G., Leipziger Vorträge, 1931.
2. Hund F., Handbuch d. Physik, XXIV, 1933.
3. Mulliken R., Rev. of Modern Physics, 1931.