Dihydrogenium

Dihydrogenium est molecula simplicissima et laevissima constructa duabus atomis hydrogenii ($\mathrm{H}{\phantom{A}}_2$).

Protypona structurae electronicae

Quamquam dihydrogenium simplicissima omnium molecularum est, eam utimur ad varias approximationes pro descriptione earum structuras electronicas explanandum. Ut theoria generalis praescribit, mos cujusque systematis quanticae (moleculae inclusae) operatore Hamiltoniano gubernatur. Hic dihydrogenii est

$\hat{H}=-\frac{{\hslash }^2}{2M}{\mathrm{\Delta }}_{{𝐑}_1}-\frac{{\hslash }^2}{2M}{\mathrm{\Delta }}_{{𝐑}_2}-\frac{{\hslash }^2}{2m}{\mathrm{\Delta }}_{{𝐫}_1}-\frac{{\hslash }^2}{2m}{\mathrm{\Delta }}_{{𝐫}_2}+\frac{e^2}{|{𝐑}_1-{𝐑}_2|}+\frac{e^2}{|{𝐫}_1-{𝐫}_2|}$

$-\frac{e^2}{|{𝐑}_1-{𝐫}_1|}-\frac{e^2}{|{𝐑}_1-{𝐫}_2|}-\frac{e^2}{|{𝐑}_2-{𝐫}_1|}-\frac{e^2}{|{𝐑}_2-{𝐫}_2|}$,

ubi $M$ — massa protonis, $m$ — massa electronis, ${𝐑}_i,{𝐫}_i$ — radii-vectores duobus nucleorum ac duobus electronum sunt ($i=1,2$). ${\mathrm{\Delta }}_{\mathit{\rho }}$ sunt operatores Laplaciani respectu componentiae radii vectoris $\mathit{\rho }=(x,y,z)$. In hac aequatione terminos cum ${\mathrm{\Delta }}_{\mathit{\rho }}$^ae repraesentant energiam kineticam nucleorum ac electronum, ceteri termini repraesentant energiae potentialia interactionum inter particulas quarum radii vectores in denominatoribus occurrunt/inveniuntur. In energia potentiali terminos positivos (cum signo "+") respondent viribus repulsivos inter particulas, negativos (cum signo "-") autem viribus attactivas (attractione).

In appropinquatione adiabatica/[[Chemia quantica#Approximatio Bornis-Oppenheimeri[1]|Bornis-Oppenheimeris]] energia kinetica nucleorum omttitur et solum Hamiltonianus electronicus:

${\hat{H}}_{el}=-\frac{{\hslash }^2}{2m}{\mathrm{\Delta }}_{{𝐫}_1}-\frac{{\hslash }^2}{2m}{\mathrm{\Delta }}_{{𝐫}_2}+\frac{e^2}{|{𝐫}_1-{𝐫}_2|}+\frac{e^2}{|{𝐑}_1-{𝐑}_2|}-\frac{e^2}{|{𝐑}_1-{𝐫}_1|}-\frac{e^2}{|{𝐑}_1-{𝐫}_2|}-\frac{e^2}{|{𝐑}_2-{𝐫}_1|}-\frac{e^2}{|{𝐑}_2-{𝐫}_2|}$,

consideratur, qui in unitatibus atomicis ($m=e=\hslash =1$) simplificatur:

${\hat{H}}_{el}=-\frac{1}{2}{\mathrm{\Delta }}_{{𝐫}_1}-\frac{1}{2}{\mathrm{\Delta }}_{{𝐫}_2}+\frac{1}{|{𝐫}_1-{𝐫}_2|}+\frac{1}{|{𝐑}_1-{𝐑}_2|}-\frac{1}{|{𝐑}_1-{𝐫}_1|}-\frac{1}{|{𝐑}_1-{𝐫}_2|}-\frac{1}{|{𝐑}_2-{𝐫}_1|}-\frac{1}{|{𝐑}_2-{𝐫}_2|}$.

Systema coordinatarum ita selecta potest, quod nucleos (protones) in punctis ${𝐑}_{1/2}={𝐑}_{l/d}=\mp (R/2,0,0)$ quiescunt (proton primus in plaga laeva, secundus in plaga dextra), itaque Hamiltonianus electronicus erit functio unicae variabilis $R$:

${\hat{H}}_{el}(R)=-\frac{1}{2}{\mathrm{\Delta }}_{{𝐫}_1}-\frac{1}{2}{\mathrm{\Delta }}_{{𝐫}_2}+\frac{1}{R}+\frac{1}{|{𝐫}_1-{𝐫}_2|}-\frac{1}{|{𝐑}_l-{𝐫}_1|}-\frac{1}{|{𝐑}_l-{𝐫}_2|}-\frac{1}{|{𝐑}_d-{𝐫}_1|}-\frac{1}{|{𝐑}_d-{𝐫}_2|}$.

In communi usu mechanicae quanticae aequatio Schrödingeri statica

${\hat{H}}_{\text{el}}(R){\mathrm{\Psi }}_{\text{el}}({𝐱}_1,{𝐱}_2|R)=E_{\text{el}}(R){\mathrm{\Psi }}_{\text{el}}({𝐱}_1,{𝐱}_2|R),$

solvenda est quoad functionem ${\mathrm{\Psi }}_{\text{el}}({𝐱}_1,{𝐱}_2|R)$ et valorem $E_{\text{el}}(R)$ proprias. Quoniam ${\hat{H}}_{el}(R)$ ab $R$ pendet, aequatio supra singula non est, sed ad quemlibet valorem $R$ solvi debet. Atque accurate aequatio solvi non potest. Enim

где $E_{\text{el}}$ —

Гамильтониан молекулы водорода симметричен относительно переменных ${𝐫}_1$ и ${𝐫}_2$, то есть не изменяется при смене нумерации электронов. Кроме того, он не зависит от спиновых переменных.

молекулы водорода разбивается на два этапа. На первом этапе рассматривается только электронная подсистема, а ядра считаются зафиксированными в точках ${𝐑}_1$ и ${𝐑}_2$.