is only their separation that affects Ei,,,l. As the nuclear charge increases, the orbital contracts in size in inverse proportion to increasing 2,forcing the electronscloser together and increasing E(cpp~Note particularly the concept of the energy level of an electron within an orbital. In any one-electron species, we may equate this with the energy of the orbital (that is, the theoretically calculated energy level of the unoccupied orbital) and with the total energy of the species, relative to completely separated electron and nucleus. Thus, to write hydrogen: IS', L, E = -1312 kJ mol-' is unambiguous; it tells us the energy of the unoccupied orhital, the energy of the electron in the orhital, and the energy of the species, while the negative of E is the energy required to remove the electron from the orhital to infinite separation. For a two-electron species, we have to specify what we mean by "energy level". T o write
u,E = -3811 kJ mol-'
helium: IsZ,
per electron
tells us the energy of the two electrons in the orbital (allowing for interelectronic repulsion), and the energy of the species relative to completely separated electrons and the He2+ nucleus. It does not tell us the energy that the orhital would have if it were unoccupied, nor the energy required to remove one electron. It is verv easv for the student to fall into the error of thinking &at thk energy level of an orhital is fixed, regardless of the number of electrons it may contain, and this misapprehension leads to much ronfusion later. Once the abure concepts have heen introduced, the same principles may he used in discussing the energies of molecular orl~italsin the hydrogen molecule. Two-Electron Molecular Systems Every student studying MO theory starts with the diacram for the hydrogen . . molecule (Fig. l). - Before attempting to interpret this diagram quantitatively, we must define what we mean by "energy level" in the case of a MO occupied by two electrons. The usual interpretation of this diagram is that i t shows the total energy of the s ~ e c i e s(in this case. the H* molecule) relative to the two skparate'atoms. By showing the energylevels of the bonding and antihonding MO's in this way, we are saying: If we bring the two atoms together and place the two electrons in n.hondine n MO. the molecule will be more stable than the sepnrate atmm hy ihearnounr ahown. Ifue place on? dpctnm in a bondmi: MOand theother in annntihrmding \lo. rhrre will beno overall c tin in energy, and no stnhle molecule iormed, herausr the ~~~
~~~~~
~
~
.
energy gained by placingone electron in the low-energy ol, MO is more thnn offset by the energy needed to place the second electron into the high-energy c',, MO. Knowing that the bond energy of the Hz molecule is 432 k J mol-', we would then say that the molecule with two electrons in the m.."orhital is a t an enerw level of -216 k J mol-' per electron, relative to the separate atoms with electrons in the atomic 1s orbitals. In contrast to the atomic examole. the ahove concept ells us nothmg about the artual rnrrgy Ie\.eI of the two eleclrons in the MO. reliltwe to Leru at infinite separation from the nuclei and from each other. The difference between molecules and atoms is, of course, that there are strong repulsive forces between the positively charged nuclei in the molecule. Hence the energy level of the two electrons within the MO must he lower than that of the total molecule by an amount equal to the repulsive interaction between the nuclei. The energy of the MO containing only one electron would then be lower still. because of the absence of interelectronic repulsion. Clearly, in order to calculate these quantities for the Hp molecule, it would be helpful to know the energy required to remove one or two electrons from the MO, in other words, the successive IE's, just as with the two-electron atomic systems. However, there is an obvious difficulty here: since the presence of the electrons is holding the nuclei together, the molecule will disintegrate if the two electrons are removed, making the measurement of the total ionization energy impossible. A different annroach is therefore needed to find AH*"for Hz2+,and the formation reaction may be broken down into a series of simple steps:
--
(1) First we make two H atoms by breaking the bond in Hdg)
(standard state):
-
Hzk)
2Hk)
AH = +432 kJ mol-' (bond energy of H,)
(2) The two H atoms are then ionized to 2H+ (all particles at infinite separation): 2H(g)
-
2Ht(g) + 2e-
AH = 2 X 1312 kJ mol-' (ionization energy of H)
(3) The two protons are then brought together to the bonding distance in the Hz molecule (74.2 pm), obviously an extremely unfavorable process.
~~~~
H+
+ H+
- H,Z~
The repulsive energy of interaction between two protons at separation r may be calculated using Coulomb's equation. On a molar basis: Eire,,= NAe2/(4m,r)
where N,
= Avogadro's
number = 6.022 X
e = charge on the proton =
loz3mol-'
1.602 X lo-'' C
c, = permittivity of vacuum = 8.854 X lo-"
C2m-' J-I
r = separation of charges = 74.2 pm = 7.42 X lo-" m
Evaluating gives: Eire,,,= 1872 kJ mol-'
atomic orbital
molecular orbitals
atomic orbital
Summing the energy terms from steps 1-3 gives: H,(g)
Figure 1. Molecular orbitals in
the hydrogen molecule
-
HZ2+(g, at 74.2 pm) + 2e-
AH = AH; for H? = 432
Volume 65
+ 2624 + 1872 = +4928 kJ mal-'
Number 5
May 1988
419
Hz2+ at 74.2 pm
fl;
...
relaxation Hz2+ at 106 pm
Figure 2. Electronic energy levels in the H. molecule, relative to separate hydrogen atoms.
(4) To return to the ground-state Hz molecule with which we start-
-
. ..
lew
1
ed, we must now place two electrons into the bonding ol, MO: H?(d
+ 2e-
HAG)
Clearly, this has AH = -4928 kJ mol-'.
add one e-2878 kJ
So for the addition of two electrons to the al, MO of Hz2+ the electronic stabilization energy associated with each is
-4928 = -2464 kJ mol-' per electron 2 relative to zero energy at infinite separation of the electrons from the protons. Knowing the energy of each electron in the orieinal Isatomicorbitals of thesenaratr atoms to be -1312 ~ -e~~~~~~ kJ mol-1, we can now easily calculate the difference in enerev between the electrons in the atomic orbitals and in the a,,.. MO giving the picture shown in Figure 2. The total enerev eained bv transferring the two electrons from the 1s ~ O ' & t h e ol, MO'S is ~
~~~
~
~
2 X 1152 = 2304 kJ mol-'
As would he expected, this is equal to the sum of the hond energy (432 k J mol-1) and the energy of repulsive nuclear interaction (1872 kJ mol-'1. The bond dissociation energy of HI is now seen in ~roportionto the other energy changes iniolved when the &oms come together. It amounts to only 1990 of the energy gained when the two electrons go from atomic to molecular orbitals, the difference being used up in bringing the oppositely charged nuclei close together. Fieure 2 reoresents the enereies of the twoelectrons within tce al, MO, taking into &count the attraction of the electrons to the nuclei and their mutual repulsion. Note that, in this figure, the antibonding a * ~MO , is not shown. It does not lie at a corresnondinp separation aboue the atomic orbitals, because "splitting" bf energy levels in this way, commonlv used in MO diagrams, is only applicable to the total mol&ular energy-leverdiagram of Figure 1. In going to a diagram showing only the electronic energy, we have moved the energies of both bonding and antibonding MO's to lower energy by an amount equal to the energy of internuclear repulsion, and the energy level of the unoccupied a * ~ . orbital cannot he evaluated from the data used in this aporoach. T o calculate the magnitude of the repulsive interaction between the two electrons in the a,, orbital of Hz, we need to know the ionization energy of rnole&ar hydrogen. However, there is a sliaht comdication here. The usually quoted value 1488 kJ mol-1, refers to the energy change of this on ionization when each species is a t its equilibrium hond a t 74.2 pm, Hz+ a t 106 p m ) A good approximaleagth (HZ tion to the energy required to ionize HZ a t constant hond lengthacan he obtained from the photo-electron spectrum; i t has the slightly greater value of 1539 kJ mol-I. Strictly, this is the transltlon from the vibrational ground slate of H2t0that vibrationally excited state of HZ' which is closest in its bond length to Hz. See, for example. Ballard. R. E. Photo-Nectron Spectroscopyand Molecular Orbital Theory, Hilger: Bristol. 1978. 420
Journal of Chemical Education
Hz* at 74 pm
. . . . . . . . . .-
Hz+at 105 pm.-
i I
iE of molecular
Figure a. Energy changes in the formation of HZ and Hzt,
The second ionization energy of Hz can therefore be found by subtracting 1539 kJ mol-1 from the total energy required to remove both electrons: overall:
H,
first IE:
Hz
second IE:
Hzf
-
+
Hz2+ 2e-
AH = 4928 kJ
Hzt (at 74 pm) + e-
AH = 1539 kJ
+ e-
AH = 3389 kJ
HgZ+(at 74 pm)
By the same argument as that used previously for the helium atom, the repulsive interaction between the two electrons in Hz is then equal to the difference between the first and second IE's of Hz, or 1850 kJ m01-I. I t is interesting to compare the value of the two energy terms in the two-electron species HZand He:
total IE of two e-, kJ 8--B- rep~lsion, kJ
H1
He
Ratio HJHe
4928
7622 2878
0.647 0.643
1850
Figure 4. Electronic energy levels in Hf at 74 pm internuclear separation.
Both the electron-nucleus attractive and electron-electron repulsive interactions increase (apparently in the same proportions) as the nuclei approach each other, reaching the limit when they coalesce to form the He nucleus and the UI, MO becomes the 1s atomic orbital. These energy changes are summarized in Figure 3. Asnotedpreviously, the energy levelof an orbital in a oneelectron system is the negative of its IE, so for the ion Hz+ we can draw a MO picture of Figure 4, giving the energy level of the electron in the q. MO. This will be the same as the energy level of the unoccupied orbital (see Fig. 4). For the Hz+ ion a t its equilibrium bond distance of 106 pm, the internuclear repulsive energy, calculated as before, is reduced to 1310 kJ mol-1, so the total energy needed to form Hz2+(at 106 pm separation) from Hz is
+
432 + (2 x 1312) 1310 = 4366 kJ mol-I If H2+has AH,' of 1488 kJ mol-I, the energy change when oneelectron is added to the bonding MO in Hz2-at 106 pm is 1488 - 4366 = -2878 kJ mol-' So for Hz+ a t its equilibrium bond distance of 106 pm, the honding MO is a t an energy level of -2878
- (-1312)
= -1566
kJ mol-'
below those of the original atomic orbitals, 33% less than the difference found a t an internuclear distance of 74 nm. The stahilizing energy is less at greater internuclear separation hecauae of the less favorable over la^ of the atomic arhitals. As Figure 3 shows, the species ~ z + - imore s stable by 51 kJ mol-' a t the greater internuclear separation because the internuclear repulsion is reduced. The stabilization produced by placing one electron in the a'. MO is less at this separation, hut this is more than compensated for by the lower internuclear repulsion. In the case of neutral Hz, the electronic stabilization energy is much greater because two electrons are in the honding MO, so a greater degree of internuclear repulsion can he overcome and the hond is shorter. The bonding energy in HZ+is the value of AH for which is equal to
AH;,")+ rJIPO,,+,- rJI;,"2*) Knowing that L Wvalues ~ of H, H+, and Hz+from Hz are, respectively, 216, (216 1312), and 1488 kJ mol-l, we can easily calculate the honding energy in Hz+ to be 256 kJ mol-'. The molecular ion Hz+ is stable with respect to dissociation into H H+. (Of course, the word "stab1e"should be interpreted with care; we cannot isolate salts containing the H2+ cation, because of its extreme reactivity, but it can be observed as a short-lived species in the gas phase.) Its bond energy is comparable to those of C-S or Cl-Cl. However, this again is only a small fraction (16%) of the energy gained by putting the electron into the bonding MO, the remainder going to overcome the proton-proton repulsion. Even one
+
+
atomic orbital
molecular orbitals
atomic Orbital
Figure 5. Energy of the species H2+ relative to (H + Ht). electron in a bonding MO (that is, a hond order of one-half) is enouah to hold two atoms toaether, because the absence of any electron-electron repulsiok wili favor such a system. For the Hz+ ion molecule, as for the neutral HZmolecule, the energy level of the vacant a * ~ orbital , cannot he found from the experimental data we are using. However, since this is a one-electron svstem. the enerw levels of the orbitals are not influenced by electron-electron repuls~onrThe energies of the MO's are therefore eaual to those cnlculated theorer~ cally on the basis of atomic&bital ~ v e r l a pAlthough .~ details of this approach are, of course, outside the scope of the present discussion, we note the result that the antihonding a*,, orbital in Hz+ is a t a level of about -1750 kJ mol-' (relative to zero a t infinity) or about -440 kJ mol-' relative to the level of the 1s AO's in the separate H atoms. In other words, the electron is in a lower energy state when simultaneously attracted to two nuclear charges, euen when it is in the antibonding MO. Note the distinction: we are specifying the energy that the single electron would have in this orbital, not the total energy of the species. If we allow for the internuclear repulsive interaction, the total energy of the species Hz+ in the antibonding electronic state (v'~.)', relative to (H Hf), becomes
...
+
and it is obviously unstable. Remembering that the total energy of the species Hz+ relative to (H H+) (that is, its honding energy) was -256 kJ mol-I, we see that the destahilization of the species produced by placing the electron in the antibonding orbital would be much greater than the stabilization produced by placing it in the bonding MO (Fig. 5). The IE of molecular Hz after relaxation of the hond length is 1488 kJ mol-'; at constant bond length this is increased to 1539 kJ mol-'. We now see that this amounts to only about 60%ofthe -2464 kJ mol-' energy level of the electron within the Hp molecule. In an orbital containing more than one electron, whether in an atom or a molecule, IE's cannot generally be used as a reliable guide to the energy levels of electrons in the orhitaL5 Nevertheless, one can find in print
+
' Slater. J. C. Quantum Theory of Molecules and Solids; McGraw-
Hill: New York. 1963: Vol. 1.
'This approach is. of course. wioely used under the title "r(oopman's tneorem' ana can gfvea reasonable approxmaton in cena n cases.
Volume 65 Number 5
May 1988
42 1
the statement that, since the IE of molecular Hz is greater than that of atomic hydrogen by 176 kJ mol-I, the bonding MO in H p is lower in energy than the 1s orbital in atomic hydrogen by this amount. The observant student might wonder why the stabilization energy of two e- in Hp is then 176 X 2 = 352 kJ mol-1, significantly less than the bond energy of 432 kJ mol-I. As we have seen, if we think of the energy of the electron in the MO, the difference between the atomic 1s and molecular q,is 1152 kJ mol-', so 176 kJ mol-' is a very poor approximation! It is tempting for the student to use this approach because it works perfectly for the one-electron, hydrogenlike species whose structure and spectra are usually studied in detail in introductory courses, but the results can be very misleading. The aim of the above discussion has been to show how a fairly detailed quantitative treatment of bonding in the hydrogen molecule may be presented to the student beginning
422
Journal of Chemical Education
the study of molecular orhital theory. General concepts are reinforced by the introduction of hard numbers based on such easily understood measurements as bond lengths and ionization energies. At the same time, a number of possible points of confusion concerning the significance of MO eneigy-level diagrams may he clarified a t the outset. In particular, we should make the distinction between the use of energy-level diagrams to show the energy of the molecular species, relative to separate atoms, and the electronic energy level of the electrons in a combined molecular orbital, relative to separate atomic orbitals. A clear understanding of these concepts will assist the student's thinking throughout further study of this important topic. Acknowledgment I am grateful to N. C . Baird for helpful discussions of this material.
