7
The H2+ Ion and Bonding
As the simplest example of covalent bonding, we consider the hydrogen molecular ion.
electron
The hydrogen molecular ion H2+ is a system composed of
two protons and a single electron. It is useful to use centre
of mass (cm) coordinates by defining the relative position
vector, R , of proton 2 with respect to proton 1, and the
position vector r of the electron relative to the centre of
mass of the two protons.
_r
_r
_r
proton 1
R
_
proton 2
The Schrodinger equation is
"
h
2 2
e2
e2
e2
h
2 2
R
r
+
(r, R) = E(r, R)
212
2e
(40 )r1 (40 )r2 (40 )R
where the reduced mass of the two-proton system is 12 = M/2, with M the proton mass, and e
is the reduced mass of the electron/two-proton system:
e =
m(2M )
'm
m + 2M
where m is the electron mass.
7.1
Born-Oppenheimer Approximation
Because nuclei are a great deal more massive than electrons, the motion of the nuclei is much
slower than that of the electrons. Thus the nuclear and electronic motions can be treated more or
less independently and it is a good approximation to determine the electronic states at any value
of R by treating the nuclei as fixed. This is the basis of the Born-Oppenheimer approximation.
In this approximation, the electron is described by an eigenfunction Uj (r, R) satisfying the Schrodinger
equation
"
h
2 2
e2
e2
e2
+
Uj (r, R) = Ej (R) Uj (r, R)
2e
(40 )r1 (40 )r2 (40 )R
This is solved keeping R constant. For each R, a set of energy eigenvalues Ej (R) and eigenfunctions Uj (r, R) is found. The functions Uj (r, R) are known as molecular orbitals.
The full wavefunction for the j th energy level at given R is taken to be the simple product
(r, R) = Fj (R) Uj (r, R)
where Fj (R) is a wavefunction describing the nuclear motion.
Substituting this form into the full Schrodinger equation and using the electronic equation yields
"
h
2 2
+ Ej (R) E Fj (R) Uj (r, R) = 0
212 R
A little vector calculus gives
2R {Fj (R) Uj (r, R)} = R R [Fj (R) Uj (r, R)]
= R Uj (r, R) R Fj (R) + Fj (R) R Uj (r, R)
= Uj (r, R) 2R Fj (R) + Fj (R) 2R Uj (r, R)
+ 2 R Uj (r, R) R Fj (R)
Assuming that the variation of the molecular orbitals with inter-proton separation, R, is weak,
we can neglect the terms involving R Uj (r, R), and 2R Uj (r, R) leaving a single-particle type
Schrodinger equation for the nuclear motion
"
h
2 2
+ Ej (R) E Fj (R) = 0
212 R
in which Ej (R) plays the role of a potential. We will return to this later.
7.2
The Electronic Ground State
We now try to investigate the lowest electronic levels of H2+ . First we look for symmetries, and
note that, since r1 = r + R/2 and r2 = r R/2, the electronic Hamiltonian is invariant under the
parity operation r r. If P denotes the parity operator, then
H]
=0
[P,
These are commuting operators, so they have can have the same eigenfunctions. These eigenfunctions are called gerade if the parity is even and ungerade if the parity is odd:
jg (r, R) = Ujg (r, R),
PU
ju (r, R) = Uju (r, R)
PU
Now think about wave functions. If R is large, the system separates into a hydrogen atom and
a proton (two degenerate states). The hydrogen atom has a large spacing between levels, so we
use degenerate perturbation theory with 1s levels only. Quite generally, this procedure of taking
linear combinations of atomic orbitals is known as the LCAO method. Note that this basis set is
normalised, but neither complete nor orthogonal.
Since there must be solutions which are eigenfunctions of the parity operator, we take normalised
linear combinations of gerade or ungerade symmetry of 1s orbitals:
g = [u1s (r1 ) + u1s (r2 )]/ 2
and
u = [u1s (r1 ) u1s (r2 )]/ 2
We calculate the expectation value of the electronic Hamiltonian using these LCAO molecular
wavefunctions:
Z
g,u
(R) =
g,u (r, R) d3 r = hu1s (r1 )|H|u
1s (r1 )i hu1s (r1 )|H|u
1s (r2 )i
g,u (r, R) H
where + and - correspond to u and g respectively, giving E g (R) and E u (R) for each value of R;
The evaluation of the integrals is complicated, but the results have the form:
E g (R) = E1s +
e2
(1 + R/a0 ) exp(2R/a0 ) + [1 (2/3)(R/a0 )2 ] exp(R/a0 )
(40 )R
1 + [1 + (R/a0 ) + (1/3)(R/a0 )2 ] exp(R/a0 )
E u (R) = E1s +
e2
(1 + R/a0 ) exp(2R/a0 ) [1 (2/3)(R/a0 )2 ] exp(R/a0 )
(40 )R
1 [1 + (R/a0 ) + (1/3)(R/a0 )2 ] exp(R/a0 )
and
where a0 is the Bohr radius and E1s is the ground-state energy of atomic hydrogen.
The two curves E g E1s and E u E1s are plotted as a function of R. Note that the curve which
corresponds to the symmetric (gerade) orbital exhibits a minimum at R = R0 , where R0 /a0 ' 2.5,
corresponding to E g E1s = 1.77 eV. Since this is an upper bound on the ground-state energy,
this implies that there is a stable bound state, a molecular ion. The curve represents an effective
attraction between the two protons. By contrast, the curve corresponding to the ungerade orbital
has no minimum, so that a H2+ ion in this state will dissociate into a proton and a hydrogen
atom. If we think of the protons being attracted by the electron and repelled by each other, the
symmetrical state should be the more tightly bound because the electron spends more of its time
between the protons, where it attracts both of them. This is an example of covalent bonding.
g,u
E 1s (eV)
4.0
3.0
E E 1s
g
2.0
E E 1s
1.0
R/a 0
0
1.0
3.0
2.0
4.0
5.0
6.0
1.0
2.0
3.0
7.3
Rotational and Vibrational Modes
We can now study the effective one-body Schrodinger equation for the nuclear motion by setting
Ej (R) = E g (R) for the ground state. Because E g (R) only depends on the magnitude of R it
represents an effective central potential, so the solutions are of the form
F g (R) =
1
RN L (R)YLMl (, )
R
where YLMl (, ) are the spherical harmonics and the function RN L (R) satisfies the radial equation
"
h
2
212
L(L + 1)
d2
+ E g (R) E RN L = 0
dR2
R2
We can approximate the centrifugal barrier term by setting it equal to its value at R = R0 , writing
h
2
Er =
L(L + 1)
212 R02
In this approximation we are treating the molecule as a rigid rotator. We can also approximate
E g (R) by Taylor expanding about R = R0 . Because this point is a minimum, the first derivative
is zero:
1
E g (R) ' E g (R0 ) + k(R R0 )2 +
2
g
where k is the value of the second derivative of E at R = R0 .
With these two approximations, the radial equation becomes
"
h
2 d2
1
+ k(R R0 )2 EN RN L = 0
2
212 dR
2
where
EN = E E g (R0 ) Er
This is the equation for a simple harmonic oscillator with energies
1
EN = h
0 (N + ),
2
N = 0, 1, 2,
where 0 = k/12 . The vibrational energies are of the order of a few tenths of an eV, whereas
the rotational energies are of the order of 103 eV. Both are much smaller than the spacing of the
electronic levels. Transitions between these various levels give rise to molecular spectra. The pure
rotational spectrum consists of closely-spaced lines in the infrared or microwave range. Transitions
which also involve changes to the vibrational state give rise to vibrational-rotational band spectra
7.4
Electronic states of the H2 Molecule
Electrons are fermions with spin 12 , so the gerade state can be double occupied, as can the ungerade state (four states in all, same as two 1s orbitals for each ion). The second electron changes
the structure of the wavefunction. Staying within LCAO, and ignoring spin, we can label basis
states as, e.g. u11s (r2 ) indicating the first electron on the second atom. The electrons are indistinguishable, so the total wavefunctions (spin times spatial) must be eigenstates of parity and the
exchange operator P12 which switches the electron labels, e.g. P12 u11s (r2 ) = u21s (r2 ). They are
fermions, hence antisymmetric: P = 1.
Assuming both electrons are 1s and in the bonding g state, and ignoring their interaction, the
LCAO 1s2 spatial wavefunction is
(r1 , r2 ) = [u1100 (r1 ) + u1100 (r2 )][u2100 (r1 ) + u2100 (r2 )]
This must be combined with a spin eigenfunction , , ( + ), or ( ), where the first
arrow represents the spin state (ms = 1) of the first electron. Since the spatial wavefunction
is symmetric under label exchange, in fact it must be combined with the antisymmetric spin
wavefunction to give the overall wavefunction in spin and space.
(r1 , r2 , s1 , s2 ) = [u1100 (r1 ) + u1100 (r2 )][u2100 (r1 ) + u2100 (r2 )][ ]
This wavefunction describes two electrons, and is non-degenerate.
The second electron also adds an electron-electron repulsion to the Hamiltonian, which can be
treated by perturbation theory.
E = h(r1 , r2 )| e2 /40 |r1 r2 | |(r1 , r2 )i
There is a lot of subtlety here, since the electrons dont interact with themselves, only with each
other, and we must avoid double-counting the interaction of 1-2 and 2-1. Well return to this in
more detail later in the context of Helium.
