4.6 The Hy­dro­gen Mol­e­c­u­lar Ion

The hy­dro­gen atom stud­ied ear­lier is where full the­o­ret­i­cal analy­sis stops. Larger sys­tems are just too dif­fi­cult to solve an­a­lyt­i­cally. Yet, it is of­ten quite pos­si­ble to un­der­stand the so­lu­tion of such sys­tems us­ing ap­prox­i­mate ar­gu­ments. As an ex­am­ple, this sec­tion con­sid­ers the H$_2^+$-​ion. This ion con­sists of two pro­tons and a sin­gle elec­tron cir­cling them. It will be shown that a chem­i­cal bond forms that holds the ion to­gether. The bond is a “co­va­lent” one, in which the pro­tons share the elec­tron.

The gen­eral ap­proach will be to com­pute the en­ergy of the ion, and to show that the en­ergy is less when the pro­tons are shar­ing the elec­tron as a mol­e­cule than when they are far apart. This must mean that the mol­e­cule is sta­ble: en­ergy must be ex­pended to take the pro­tons apart.

The ap­prox­i­mate tech­nique to be used to find the state of low­est en­ergy is a ba­sic ex­am­ple of what is called a “vari­a­tional method.”

4.6.1 The Hamil­ton­ian

First the Hamil­ton­ian is needed. Since the pro­tons are so much heav­ier than the elec­tron, to good ap­prox­i­ma­tion they can be con­sid­ered fixed points in the en­ergy com­pu­ta­tion. That is called the “Born-Op­pen­heimer ap­prox­i­ma­tion”. In this ap­prox­i­ma­tion, only the Hamil­ton­ian of the elec­tron is needed. It makes things a lot sim­pler, which is why the Born-Op­pen­heimer ap­prox­i­ma­tion is a com­mon as­sump­tion in ap­pli­ca­tions of quan­tum me­chan­ics.

Com­pared to the Hamil­ton­ian of the hy­dro­gen atom of sec­tion 4.3.1, there are now two terms to the po­ten­tial en­ergy, the elec­tron ex­pe­ri­enc­ing at­trac­tion to both pro­tons:

$$H = -\frac{\hbar^2}{2m_{\rm e}}\nabla^2 - \frac{e^2}{4\pi\... ..._{\rm {l}}} - \frac{e^2}{4\pi\epsilon_0}\frac{1}{r_{\rm {r}}}$$ (4.74)

where $r_{\rm {l}}$ and $r_{\rm {r}}$ are the dis­tances from the elec­tron to the left and right pro­tons,

$$r_{\rm {l}}\equiv \vert{\skew0\vec r}- {\skew0\vec r}_{\rm ... ...{r}}\equiv \vert{\skew0\vec r}- {\skew0\vec r}_{\rm {rp}}\vert$$ (4.75)

with ${\skew0\vec r}_{\rm {lp}}$ the po­si­tion of the left pro­ton and ${\skew0\vec r}_{\rm {rp}}$ that of the right one.

The hy­dro­gen ion in the Born-Op­pen­heimer ap­prox­i­ma­tion can be solved an­a­lyt­i­cally us­ing pro­late spher­oidal co­or­di­nates. How­ever, ap­prox­i­ma­tions will be used here. For one thing, you learn more about the physics that way.

Key Points

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In the Born-Op­pen­heimer ap­prox­i­ma­tion, the elec­tronic struc­ture is com­puted as­sum­ing that the nu­clei are at fixed po­si­tions.

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The Hamil­ton­ian in the Born-Op­pen­heimer ap­prox­i­ma­tion has been found. It is above.

4.6.2 En­ergy when fully dis­so­ci­ated

The fully dis­so­ci­ated state is when the pro­tons are very far apart and there is no co­her­ent mol­e­cule, as in fig­ure 4.14. The best the elec­tron can do un­der those cir­cum­stances is to com­bine with ei­ther pro­ton, say the left one, and form a hy­dro­gen atom in the ground state of low­est en­ergy. In that case the right pro­ton will be alone.

Fig­ure 4.14: Hy­dro­gen atom plus free pro­ton far apart.

Ac­cord­ing to the so­lu­tion for the hy­dro­gen atom, the elec­tron loses 13.6 eV of en­ergy by go­ing in the ground state around the left pro­ton. Of course, it would lose the same en­ergy go­ing into the ground state around the right pro­ton, but for now, as­sume that it is around the left pro­ton.

The wave func­tion de­scrib­ing this state is just the ground state $\psi_{100}$ de­rived for the hy­dro­gen atom, equa­tion (4.40), but the dis­tance should be mea­sured from the po­si­tion ${\skew0\vec r}_{\rm {lp}}$ of the left pro­ton in­stead of from the ori­gin:

$$\psi=\psi_{100}(\vert{\skew0\vec r}- {\skew0\vec r}_{\rm {lp}}\vert)$$

To shorten the no­ta­tions, this wave func­tion will be de­noted by $\psi_{\rm {l}}$:

$$\psi_{\rm {l}}({\skew0\vec r}) \equiv \psi_{100}(\vert{\skew0\vec r}- {\skew0\vec r}_{\rm {lp}}\vert)$$ (4.76)

Sim­i­larly the wave func­tion that would de­scribe the elec­tron as be­ing in the ground state around the right pro­ton will be de­noted as $\psi_{\rm {r}}$, with

$$\psi_{\rm {r}}({\skew0\vec r}) \equiv \psi_{100}(\vert{\skew0\vec r}- {\skew0\vec r}_{\rm {rp}}\vert)$$ (4.77)

Key Points

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When the pro­tons are far apart, there are two low­est en­ergy states, $\psi_{\rm {l}}$ and $\psi_{\rm {r}}$, in which the elec­tron is in the ground state around the left, re­spec­tively right, pro­ton. In ei­ther case there is an hy­dro­gen atom plus a free pro­ton.

4.6.3 En­ergy when closer to­gether

Fig­ure 4.15: Hy­dro­gen atom plus free pro­ton closer to­gether.

When the pro­tons get a bit closer to each other, but still well apart, the dis­tance $r_{\rm {r}}$ be­tween the elec­tron or­bit­ing the left pro­ton and the right pro­ton de­creases, as sketched in fig­ure 4.15. The po­ten­tial that the elec­tron sees is now not just that of the left pro­ton; the dis­tance $r_{\rm {r}}$ is no longer so large that the $\vphantom{0}\raisebox{1.5pt}{$-$}$$e^2$$\raisebox{.5pt}{$/$}$​$4\pi\epsilon_0r_{\rm {r}}$ po­ten­tial can be com­pletely ne­glected.

How­ever, as­sum­ing that the right pro­ton stays suf­fi­ciently clear of the elec­tron wave func­tion, the dis­tance $r_{\rm {r}}$ be­tween elec­tron and right pro­ton can still be av­er­aged out as be­ing the same as the dis­tance $d$ be­tween the two pro­tons. Within that ap­prox­i­ma­tion, it sim­ply adds the con­stant $\vphantom{0}\raisebox{1.5pt}{$-$}$$e^2$$\raisebox{.5pt}{$/$}$​$4\pi\epsilon_0d$ to the Hamil­ton­ian of the elec­tron. And adding a con­stant to a Hamil­ton­ian does not change the eigen­func­tion; it only changes the eigen­value, the en­ergy, by that con­stant. So the ground state $\psi_{\rm {l}}$ of the left pro­ton re­mains a good ap­prox­i­ma­tion to the low­est en­ergy wave func­tion.

More­over, the de­crease in en­ergy due to the elec­tron/right pro­ton at­trac­tion is bal­anced by an in­crease in en­ergy of the pro­tons by their mu­tual re­pul­sion, so the to­tal en­ergy of the ion re­mains the same. In other words, the right pro­ton is to first ap­prox­i­ma­tion nei­ther at­tracted nor re­pelled by the neu­tral hy­dro­gen atom on the left. To sec­ond ap­prox­i­ma­tion the right pro­ton does change the wave func­tion of the elec­tron a bit, re­sult­ing in some at­trac­tion, but this ef­fect will be ig­nored.

So far, it has been as­sumed that the elec­tron is cir­cling the left pro­ton. But the case that the elec­tron is cir­cling the right pro­ton is of course phys­i­cally equiv­a­lent. In par­tic­u­lar the en­ergy must be ex­actly the same by sym­me­try.

Key Points

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To first ap­prox­i­ma­tion, there is no at­trac­tion be­tween the free pro­ton and the neu­tral hy­dro­gen atom, even some­what closer to­gether.

4.6.4 States that share the elec­tron

The ap­prox­i­mate en­ergy eigen­func­tion $\psi_{\rm {l}}$ that de­scribes the elec­tron as be­ing around the left pro­ton has the same en­ergy as the eigen­func­tion $\psi_{\rm {r}}$ that de­scribes the elec­tron as be­ing around the right one. There­fore any lin­ear com­bi­na­tion of the two,

$$\psi = a \psi_{\rm {l}} + b \psi_{\rm {r}}$$ (4.78)

is also an eigen­func­tion with the same en­ergy. In such com­bi­na­tions, the elec­tron is shared by the pro­tons, in ways that de­pend on the cho­sen val­ues of $a$ and $b$.

Note that the con­stants $a$ and $b$ are not in­de­pen­dent: the wave func­tion should be nor­mal­ized, $\langle\psi\vert\psi\rangle$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1. Since $\psi_{\rm {l}}$ and $\psi_{\rm {r}}$ are al­ready nor­mal­ized, and as­sum­ing that $a$ and $b$ are real, this works out to

$$\langle a \psi_{\rm {l}} + b \psi_{\rm {r}}\vert a \psi_{\... ...+ 2 ab \langle\psi_{\rm {l}}\vert \psi_{\rm {r}}\rangle = 1 %$$ (4.79)

As a con­se­quence, only the ra­tio the co­ef­fi­cients $a$$\raisebox{.5pt}{$/$}$​$b$ can be cho­sen freely.

A par­tic­u­larly in­ter­est­ing case is the an­ti­sym­met­ric one, $b$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$$a$. As fig­ure 4.16 shows, in this state there is zero prob­a­bil­ity of find­ing the elec­tron at the sym­me­try plane mid­way in be­tween the pro­tons.

Fig­ure 4.16: The elec­tron be­ing an­ti­sym­met­ri­cally shared.

The rea­son is that $\psi_{\rm {l}}$ and $\psi_{\rm {r}}$ are equal at the sym­me­try plane, mak­ing their dif­fer­ence zero.

This is ac­tu­ally a quite weird re­sult. You com­bine two states, in both of which the elec­tron has some prob­a­bil­ity of be­ing at the sym­me­try plane, and in the com­bi­na­tion the elec­tron has zero prob­a­bil­ity of be­ing there. The prob­a­bil­ity of find­ing the elec­tron at any po­si­tion, in­clud­ing the sym­me­try plane, in the first state is given by $\vert\psi_{\rm {l}}\vert^2$. Sim­i­larly, the prob­a­bil­ity of find­ing the elec­tron in the sec­ond state is given by $\vert\psi_{\rm {r}}\vert^2$. But for the com­bined state na­ture does not do the log­i­cal thing of adding the two prob­a­bil­i­ties to­gether to come up with $\frac12\vert\psi_{\rm {l}}\vert^2+\frac12\vert\psi_{\rm {r}}\vert^2$.

In­stead of adding phys­i­cally ob­serv­able prob­a­bil­i­ties, na­ture squares the un­ob­serv­able wave func­tion $a\psi_{\rm {l}}-a\psi_{\rm {r}}$ to find the new prob­a­bil­ity dis­tri­b­u­tion. The squar­ing adds a cross term, $-2a^2\psi_{\rm {l}}\psi_{\rm {r}}$, that sim­ply adding prob­a­bil­i­ties does not have. This term has the phys­i­cal ef­fect of pre­vent­ing the elec­tron to be at the sym­me­try plane, but it does not have a nor­mal phys­i­cal ex­pla­na­tion. There is no force re­pelling the elec­trons from the sym­me­try plane or any­thing like that. Yet it looks as if there is one in this state.

The most im­por­tant com­bi­na­tion of $\psi_{\rm {l}}$ and $\psi_{\rm {r}}$ is the sym­met­ric one, $b$ $\vphantom0\raisebox{1.5pt}{$=$}$ $a$. The ap­prox­i­mate wave func­tion then takes the form $a(\psi_{\rm {l}}+\psi_{\rm {r}})$. That can be writ­ten out fully in terms of the hy­dro­gen ground state wave func­tion as:

$$\fbox{$\displaystyle \Psi \approx a \left[ \psi_{100}(\v... ...i_{100}(r) \equiv \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0} $} %$$ (4.80)

where $a_0$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.53 Å is the Bohr ra­dius and ${\skew0\vec r}$, ${\skew0\vec r}_{\rm {lp}}$, and ${\skew0\vec r}_{\rm {rp}}$ are again the po­si­tion vec­tors of elec­tron and pro­tons. In this case, there is in­creased prob­a­bil­ity for the elec­tron to be at the sym­me­try plane, as shown in fig­ure 4.17.

Fig­ure 4.17: The elec­tron be­ing sym­met­ri­cally shared.

A state in which the elec­tron is shared is truly a case of the elec­tron be­ing in two dif­fer­ent places at the same time. For if in­stead of shar­ing the elec­tron, each pro­ton would be given its own half elec­tron, the ex­pres­sion for the Bohr ra­dius, $a_0$ $\vphantom0\raisebox{1.5pt}{$=$}$ $4\pi\epsilon_0\hbar^2$$\raisebox{.5pt}{$/$}$​${m_{\rm e}}e^2$, shows that the eigen­func­tions $\psi_{\rm {l}}$ and $\psi_{\rm {r}}$ would have to blow up in ra­dius by a fac­tor four. (Be­cause of $m_{\rm e}$ and $e$; the sec­ond fac­tor $e$ is the pro­ton charge.) The en­ergy would then re­duce by the same fac­tor four. That is sim­ply not what hap­pens. You get the physics of a com­plete elec­tron be­ing present around each pro­ton with 50% prob­a­bil­ity, not the physics of half an elec­tron be­ing present for sure.

Key Points

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This sub­sec­tion brought home the phys­i­cal weird­ness aris­ing from the math­e­mat­ics of the un­ob­serv­able wave func­tion.

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In par­tic­u­lar, within the ap­prox­i­ma­tions made, there ex­ist states that all have the same ground state en­ergy, but whose phys­i­cal prop­er­ties are dra­mat­i­cally dif­fer­ent.

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The pro­tons may share the elec­tron. In such states there is a prob­a­bil­ity of find­ing the elec­tron around ei­ther pro­ton.

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Even if the pro­tons share the elec­tron equally as far as the prob­a­bil­ity dis­tri­b­u­tion is con­cerned, dif­fer­ent phys­i­cal states are still pos­si­ble. In the sym­met­ric case that the wave func­tions around the pro­tons have the same sign, there is in­creased prob­a­bil­ity of the elec­tron be­ing found in be­tween the pro­tons. In the an­ti­sym­met­ric case of op­po­site sign, there is de­creased prob­a­bil­ity of the elec­tron be­ing found in be­tween the pro­tons.

4.6.5 Com­par­a­tive en­er­gies of the states

The pre­vi­ous two sub­sec­tions de­scribed states of the hy­dro­gen mol­e­c­u­lar ion in which the elec­tron is around a sin­gle pro­ton, as well as states in which it is shared be­tween pro­tons. To the ap­prox­i­ma­tions made, all these states have the same en­ergy. Yet, if the ex­pec­ta­tion en­ergy of the states is more ac­cu­rately ex­am­ined, it turns out that in­creas­ingly large dif­fer­ences show up when the pro­tons get closer to­gether. The sym­met­ric state has the least en­ergy, the an­ti­sym­met­ric state the high­est, and the states where the elec­tron is around a sin­gle pro­ton have some­thing in be­tween.

It is not that easy to see phys­i­cally why the sym­met­ric state has the low­est en­ergy. An ar­gu­ment is of­ten made that in the sym­met­ric case, the elec­tron has in­creased prob­a­bil­ity of be­ing in be­tween the pro­tons, where it is most ef­fec­tive in pulling them to­gether. How­ever, ac­tu­ally the po­ten­tial en­ergy of the sym­met­ric state is higher than for the other states: putting the elec­tron mid­way in be­tween the two pro­tons means hav­ing to pull it away from one of them.

The Feyn­man lec­tures on physics, [22], ar­gue in­stead that in the sym­met­ric case, the elec­tron is some­what less con­strained in po­si­tion. Ac­cord­ing to the Heisen­berg un­cer­tainty re­la­tion­ship, that al­lows it to have less vari­a­tion in mo­men­tum, hence less ki­netic en­ergy. In­deed the sym­met­ric state does have less ki­netic en­ergy, but this is al­most to­tally achieved at the cost of a cor­re­spond­ing in­crease in po­ten­tial en­ergy, rather than due to a larger area to move in at the same po­ten­tial en­ergy. And the ki­netic en­ergy is not re­ally di­rectly re­lated to avail­able area in any case. The ar­gu­ment is not in­cor­rect, but in what sense it ex­plains, rather than just sum­ma­rizes, the an­swer is de­bat­able.

Key Points

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The en­er­gies of the dis­cussed states are not the same when ex­am­ined more closely.

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The sym­met­ric state has the low­est en­ergy, the an­ti­sym­met­ric one the high­est.

4.6.6 Vari­a­tional ap­prox­i­ma­tion of the ground state

The ob­jec­tive of this sub­sec­tion is to use the rough ap­prox­i­ma­tions of the pre­vi­ous sub­sec­tions to get some very con­crete data on the hy­dro­gen mol­e­c­u­lar ion.

The idea is sim­ple but pow­er­ful: since the true ground state is the state of low­est en­ergy among all wave func­tions, the best among ap­prox­i­mate wave func­tions is the one with the low­est en­ergy. In the pre­vi­ous sub­sec­tions, ap­prox­i­ma­tions to the ground state were dis­cussed that took the form $a\psi_{\rm {l}}+b\psi_{\rm {r}}$, where $\psi_{\rm {l}}$ de­scribed the state where the elec­tron was in the ground state around the left pro­ton, and $\psi_{\rm {r}}$ where it was around the right pro­ton. The wave func­tion of this type with the low­est en­ergy will pro­duce the best pos­si­ble data on the true ground state, {N.6}.

Note that all that can be changed in the ap­prox­i­ma­tion $a\psi_{\rm {l}}+b\psi_{\rm {r}}$ to the wave func­tion is the ra­tio of the co­ef­fi­cients $a$$\raisebox{.5pt}{$/$}$​$b$, and the dis­tance be­tween the pro­tons $d$. If the ra­tio $a$$\raisebox{.5pt}{$/$}$​$b$ is fixed, $a$ and $b$ can be com­puted from it us­ing the nor­mal­iza­tion con­di­tion (4.79), so there is no free­dom to chose them in­di­vid­u­ally. The ba­sic idea is now to search through all pos­si­ble val­ues of $a$$\raisebox{.5pt}{$/$}$​$b$ and $d$ un­til you find the val­ues that give the low­est en­ergy.

This sort of method is called a “vari­a­tional method” be­cause at the min­i­mum of en­ergy, the de­riv­a­tives of the en­ergy must be zero. That in turn means that the en­ergy does not vary with in­fin­i­tes­i­mally small changes in the pa­ra­me­ters $a$$\raisebox{.5pt}{$/$}$​$b$ and $d$.

To find the min­i­mum en­ergy is noth­ing that an en­gi­neer­ing grad­u­ate stu­dent could not do, but it does take some ef­fort. You can­not find the best val­ues of $a$$\raisebox{.5pt}{$/$}$​$b$ and $d$ an­a­lyt­i­cally; you have to have a com­puter find the en­ergy at a lot of val­ues of $d$ and $a$$\raisebox{.5pt}{$/$}$​$b$ and search through them to find the low­est en­ergy. Or ac­tu­ally, sim­ply hav­ing a com­puter print out a ta­ble of val­ues of en­ergy ver­sus $d$ for a few typ­i­cal val­ues of $a$$\raisebox{.5pt}{$/$}$​$b$, in­clud­ing $a$$\raisebox{.5pt}{$/$}$​$b$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $a$$\raisebox{.5pt}{$/$}$​$b$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$1, and look­ing at the print-out to see where the en­ergy is most neg­a­tive works fine too. That is what the num­bers be­low came from.

You do want to eval­u­ate the en­ergy of the ap­prox­i­mate states ac­cu­rately as the ex­pec­ta­tion value. If you do not find the en­ergy as the ex­pec­ta­tion value, the re­sults may be less de­pend­able. For­tu­nately, find­ing the ex­pec­ta­tion en­ergy for the given ap­prox­i­mate wave func­tions can be done ex­actly; the de­tails are in de­riva­tion {D.21}.

If you ac­tu­ally go through the steps, your print-out should show that the min­i­mum en­ergy oc­curs when $a$ $\vphantom0\raisebox{1.5pt}{$=$}$ $b$, the sym­met­ric state, and at a sep­a­ra­tion dis­tance be­tween the pro­tons equal to about 1.3 Å. This sep­a­ra­tion dis­tance is called the “bond length”. The min­i­mum en­ergy is found to be about 1.8 eV be­low the en­ergy of -13.6 eV when the pro­tons are far apart. So it will take at least 1.8 eV to take the ground state with the pro­tons at a dis­tance of 1.3 Å com­pletely apart into well sep­a­rated pro­tons. For that rea­son, the 1.8 eV is called the “bind­ing en­ergy”.

Key Points

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The best ap­prox­i­ma­tion to the ground state us­ing ap­prox­i­mate wave func­tions is the one with the low­est en­ergy.

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Mak­ing such an ap­prox­i­ma­tion is called a vari­a­tional method.

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The en­ergy should be eval­u­ated as the ex­pec­ta­tion value of the Hamil­ton­ian.

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Us­ing com­bi­na­tions of $\psi_{\rm {l}}$ and $\psi_{\rm {r}}$ as ap­prox­i­mate wave func­tions, the ap­prox­i­mate ground state turns out to be the one in which the elec­tron is sym­met­ri­cally shared be­tween the pro­tons.

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The bind­ing en­ergy is the en­ergy re­quired to take the mol­e­cule apart.

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The bond length is the dis­tance be­tween the nu­clei.

4.6.6 Re­view Ques­tions

1.

The so­lu­tion for the hy­dro­gen mol­e­c­u­lar ion re­quires elab­o­rate eval­u­a­tions of in­ner prod­uct in­te­grals and a com­puter eval­u­a­tion of the state of low­est en­ergy. As a much sim­pler ex­am­ple, you can try out the vari­a­tional method on the one-di­men­sion­al case of a par­ti­cle stuck in­side a pipe, as dis­cussed in chap­ter 3.5. Take the ap­prox­i­mate wave func­tion to be:

$$\psi = a x(\ell -x)$$

Find $a$ from the nor­mal­iza­tion re­quire­ment that the to­tal prob­a­bil­ity of find­ing the par­ti­cle in­te­grated over all pos­si­ble $x$ po­si­tions is one. Then eval­u­ate the en­ergy $\langle{E}\rangle$ as $\langle\psi\vert H\vert\psi\rangle$, where ac­cord­ing to chap­ter 3.5.3, the Hamil­ton­ian is

$$H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}$$

Com­pare the ground state en­ergy with the ex­act value,

$$E_1=\hbar^2\pi^2/2m\ell^2$$

(Hints: $\int_0^{\ell}x(\ell -x){ \rm d}{x}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell^3$$\raisebox{.5pt}{$/$}$​6 and $\int_0^{\ell}x^2(\ell -x)^2{ \rm d}{x}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell^5$$\raisebox{.5pt}{$/$}$​30)

So­lu­tion hione-a

4.6.7 Com­par­i­son with the ex­act ground state

The vari­a­tional so­lu­tion de­rived in the pre­vi­ous sub­sec­tion is only a crude ap­prox­i­ma­tion of the true ground state of the hy­dro­gen mol­e­c­u­lar ion. In par­tic­u­lar, the as­sump­tion that the mol­e­c­u­lar wave func­tion can be ap­prox­i­mated us­ing the in­di­vid­ual atom ground states is only valid when the pro­tons are far apart, and is in­ac­cu­rate if they are 1.3 Å apart, as the so­lu­tion says they are.

Yet, for such a poor wave func­tion, the main re­sults are sur­pris­ingly good. For one thing, it leaves no doubt that a bound state re­ally ex­ists. The rea­son is that the true ground state must al­ways have a lower en­ergy than any ap­prox­i­mate one. So, the bind­ing en­ergy must be at least the 1.8 eV pre­dicted by the ap­prox­i­ma­tion.

In fact, the ex­per­i­men­tal bind­ing en­ergy is 2.8 eV. The found ap­prox­i­mate value is only a third less, pretty good for such a sim­plis­tic as­sump­tion for the wave func­tion. It is re­ally even bet­ter than that, since a fair com­par­i­son re­quires the ab­solute en­er­gies to be com­pared, rather than just the bind­ing en­ergy; the ap­prox­i­mate so­lu­tion has $\vphantom{0}\raisebox{1.5pt}{$-$}$15.4 eV, rather than $\vphantom{0}\raisebox{1.5pt}{$-$}$16.4. This high ac­cu­racy for the en­ergy us­ing only mar­ginal wave func­tions is one of the ad­van­tages of vari­a­tional meth­ods {A.7}.

The es­ti­mated bond length is not too bad ei­ther; ex­per­i­men­tally the pro­tons are 1.06 Å apart in­stead of 1.3 Å. (The an­a­lyt­i­cal so­lu­tion us­ing spher­oidal co­or­di­nates men­tioned ear­lier gives 2.79 eV and 1.06 Å, in good agree­ment with the ex­per­i­men­tal val­ues. But even that so­lu­tion is not re­ally ex­act: the elec­tron does not bind the nu­clei to­gether rigidly, but more like a spring force. As a re­sult, the nu­clei be­have like a har­monic os­cil­la­tor around their com­mon cen­ter of grav­ity. Even in the ground state, they will re­tain some un­cer­tainty around the 1.06 Å po­si­tion of min­i­mal en­ergy, and a cor­re­spond­ing small amount of ad­di­tional mol­e­c­u­lar ki­netic and po­ten­tial en­ergy. The im­proved Born-Op­pen­heimer ap­prox­i­ma­tion of chap­ter 9.2.3 can be used to com­pute such ef­fects.)

The qual­i­ta­tive prop­er­ties of the ap­prox­i­mate wave func­tion are cor­rect. For ex­am­ple, it can be seen that the ex­act ground state wave func­tion must be real and pos­i­tive {A.8}; the ap­prox­i­mate wave func­tion is real and pos­i­tive too.

It can also be seen that the ex­act ground state must be sym­met­ric around the sym­me­try plane mid­way be­tween the pro­tons, and ro­ta­tion­ally sym­met­ric around the line con­nect­ing the pro­tons, {A.9}. The ap­prox­i­mate wave func­tion has both those prop­er­ties too.

In­ci­den­tally, the fact that the ground state wave func­tion must be real and pos­i­tive is a much more solid rea­son that the pro­tons must share the elec­tron sym­met­ri­cally than the phys­i­cal ar­gu­ments given in sub­sec­tion 4.6.5, even though it is more math­e­mat­i­cal.

Key Points

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The ob­tained ap­prox­i­mate ground state is pretty good.

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The pro­tons re­ally share the elec­tron sym­met­ri­cally in the ground state.