Chemia quantica

Formula:Vicificanda Chemia quantica est pars chemiae physicaeque theoreticarum tractans structuram ac proprietates moleculārum prŏfĭciscendo ex mechanicā quanticā electronum in campō electricō heterogeneō nucleis atomicis inducto moventium.

Primi chemici quantici

erant

• Fritz London (f)
• Walter Heitler (f)
• Hermann Weyl (m,p)
• Yuri Rumer

ab

• Vladimirus_Fock (f)
• Hans Hellmann (f)

sequuntur.

Notiones chemiae quanticae

• functio undarum/undina electronica (multae electronum)
• densitas electronica et matrix densitatis electronicae
• coordinates atomici (nucleari)
• orbitalĭa molecularia (orbital moleculare)
• Hamiltonianum (operatrum) vel Hamiltoniana (matrix)
• Fockianum (operatrum) vel Fockiana (matrix)
• status infimus vel imus
• mulciplicitas status
• potential
• pseudopotential
• corculum
• (hyper)superficies energiae potentialis vel superficies energetica

Formalismus

Aequatio Schrödingeris statica

Formaliter statūs cujusquid systematis corpōrum legibus mechanicae quanticae descriptōrum functione undariā $\psi ({𝐱}_1,{𝐱}_2,\dots ,{𝐱}_N,t)$ exprĭmĭtur. Hujus functionis argumenta (variabiles) sunt omnium systematis corpōrum coordinatae ${𝐱}_1,{𝐱}_2,\dots ,{𝐱}_N$ et tempus $t$. Functiones undariae legi Schrödingeri: $$i\hslash \frac{\partial }{\partial t}\psi =H\psi$$subordinantur, ubi operator Hamiltonianus $H$ (in chemia quantica) a tempore non pendet. Ponatur $\psi (\dots ,t)=\psi (\dots )\exp (-\frac{iEt}{\hslash })$, ubi $E$ quidam numerus realis est, pars prima (laeva) suprae aequationis Schrödingeri erit $E\psi (\dots )\exp (-\frac{iEt}{\hslash })$, et ambae aequationis partes, multiplicatore exponentiale divisis, dabunt $$H\psi (\dots )=E\psi (\dots )$$

ubi tempus ($t$) ab aliis variabilibus separatur. Haec aequatio, ad functionem energiamque, aequatio Schrödingeri statica nuncupatur, quae aspectum problematis de vectoribus ac valoribus propriis habet, continet in latere laevō operatorem differentialem Hamiltonianem $H$ (vide infra) ergo per functionem undariam dīvĭdi non posset. Aequatio statica multam solutionem habet quaevis quārum quemdam status quanticum systematis Hamiltoniano $H$ descriptis praebet. Functio autem undaria solō a variabilibus ${𝐱}_1,{𝐱}_2,\dots ,{𝐱}_N$ pendet.

Operatrum Hamiltonianum moleculare

In circumstantiā non-relativistica Hamiltonianum cujusquid systematis atomārum, id est systematis nucleōrum ac electronum viribus electricis interagentinbus, est: $$H=T_n+T_e+V_{nn}+V_{ne}+V_{ee}$$ubi $T_n$ energia kinetica nuleōrum (suum operatrum): $$T_n=-\frac{{\hslash }^2}{2}\sum _{\alpha =1}^A\frac{1}{M_{\alpha }}\underset{\alpha }{\overset{2}{\mathrm{\nabla }}}$$est (hic $A$ numerus nucleōrum est, $M_{\alpha }$ massa $\alpha$-isimi nuclei est); similiter $T_e$ energia kinetica electronum: $$T_e=-\frac{{\hslash }^2}{2m_e}\sum _{i=1}^N\underset{i}{\overset{2}{\mathrm{\nabla }}}$$est ($m_e$ est massa electronis). Enim operatora $V_{nn},V_{ne},V_{ee}$ sunt quodque energiae potentialis interactionis electrostaticae inter nucleis $$V_{nn}=\frac{{\mathrm{e}}^{\mathrm{2}}}{2}\sum _{\alpha \ne \beta }\frac{Z_{\alpha }{Z}_{\beta }}{|{𝐑}_{\alpha }-{𝐑}_{\beta }|}$$ (hic $Z_{\alpha }$ onus electricus $\alpha$-isimi nuclei in unitatibus oneris electronis $\mathrm{e}$ est - ergo aliqui numerus naturalis minor quam 118, ${𝐑}_{\alpha }$ vector tri-dimensionalis loci $\alpha$-isimi nuclei est); energiae potentialis ineractionis electrostaticae inter nucleis et electronibus: $$V_{ne}=-{\mathrm{e}}^{\mathrm{2}}\sum _{\alpha j}\frac{Z_{\alpha }}{|{𝐑}_{\alpha }-{𝐫}_j|}$$(hic ${𝐫}_j$ vector loci $j$-simi electronis est); energiae potentialis interactionis electronum: $$V_{ee}=\frac{{\mathrm{e}}^{\mathrm{2}}}{2}\sum _{i\ne j}\frac{1}{|{𝐫}_i-{𝐫}_j|}.$$Notandum interdum est, quod variabiles ad electrona pertinentes ${𝐱}_j$, a quibus functio undaria pendet, non solō tri-dimensionales locuum vectores ${𝐫}_j=(r_{xj},r_{yj},r_{zj})$ sunt, sed autem variabiles electronum spirulares $s_j$, quae unum ambōrum valōrum ±½ admissibilium pro fermione spirulitatis vel volubilitatis ½ accipit: ${𝐱}_j=({𝐫}_j,s_j)$. Debitā autem consideratione adhibitā notare possumus Hamiltonianum a variabilibus spirularibus non pendere. Secundum leges symmetriae solutiones problematis de vectoribus propriis cum tale Hamiltoniano erunt simul vectores (functiones undariae) propriae operatoris totae spirulitates vel multiplitatis spirulare (vide infra).

Approximatio Bornis-Oppenheimeri

Comparando energias kineticas nucleorum et electronum notamus massam levissimi nuclei - hic hydrogenii ca. 2000-ies gravior esse electrono. Ergo energia kinetica nucleorum $T_n$ magno minor est quam energia kinetica electronum $T_e$ ita ut illa neglecta esse possit. Ita familiam Hamiltonianōrum electronicōrum habemus $$H_e=T_e+V_{nn}+V_{ne}+V_{ee}$$cujus membra copiā locōrum nucleōrum $\{{𝐑}_{\alpha }|\alpha =1,\dots ,A\}$ distinguuntur. Physice, autem, quaeque data configuratio nucleōrum $\{{𝐑}_{\alpha }|\alpha =1,\dots ,A\}$ inducit campum vel potential electricum in quō electrones moventur; nuclei autem, quia ipsōrum energia kinetica zeri aequalis posita est, in statu suo quiescendi persĕvērant. Motūs electronum vicissim functione undariā solo electronum ${\mathrm{\Psi }}_e({𝐱}_1,{𝐱}_2,\dots ,{𝐱}_N)$, eandem functio undaria electronica nōmĭnēmus, exprĭmuntur. Quoniam Hamiltonianum electronicum pro quōque configuratione nucleōrum $\{{𝐑}_{\alpha }\}$ potentialem electricam congruentem habet, functio undaria electronica aequationis Schödingeri staticae sătisfăciens aeque ab configuratione $\{{𝐑}_{\alpha }\}$ pendet; ergo scribere possumus ${\mathrm{\Psi }}_e={\mathrm{\Psi }}_e(\{{𝐱}_j\}\mid \{{𝐑}_{\alpha }\})$ ad dependentiam functionis electronicae ab configuratione $\{{𝐑}_{\alpha }\}$ denotandum. Enimvero energia tota electronica quid est valor medius exspectatus Hamiltoniani electronici $E_e=⟨{\mathrm{\Psi }}_e|H_e|{\mathrm{\Psi }}_e{⟩}_{𝐱}$ est autem finctio configurationis $E_e=E_e(\{{𝐑}_{\alpha }\})$ exprimens energiam potentialem, gubernantem motum nucleōrum, si eōrum energia kinetica resumpta sit.

Structura atomi

Atomi sunt systemata simplicissima ex electronibus ac nucleis compositae quia quaeque atomus solum nucleum habet/continet. Structura atomi modō mechanicae quanticae describeri postest.

Atomus Hydrogenii

Ponendo nucleum requiscere, aequationem Schrödingeri ad atomum simplicissimam - illā hydrogenii, sed cum onere nuclei $Z$ (numerō positivō integrō) compendii gratiā scribere possumus:

$$\hat{H}\psi (𝐫)=(-\frac{{\hslash }^2{\mathrm{\nabla }}^2}{2m_{\mathrm{e}}}-\frac{Z{\mathrm{e}}^2}{\left|𝐫\right|})\psi (𝐫)=E\psi (𝐫)$$

quid in unitatibus atomicis transcribi potest $$\hat{H}\psi (𝐫)=(-\frac{{\mathrm{\nabla }}^2}{2}-\frac{Z}{\left|𝐫\right|})\psi (𝐫)=E\psi (𝐫).$$Ut eam resolvāmus a coordinatis cartesianis ad coordinatas sphaericas $$𝐫=(x,y,z)=r(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$$transeāmus, in quibus operator Laplacianus in coordinatis Cartesianis formam: $${\mathrm{\nabla }}^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$$ habens, accipit autem formam$${\mathrm{\nabla }}^2=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial }{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2}{\partial {\phi }^2}.$$Quae autem simplicius videtur quia cum debita dexteritate adhibenda functio undaria electronica in campō nucleō onere $Z$ inductō in productō partium duārum/factōrum duōrum dissolvi potest: $${\psi }_{n\ell m}(r,\theta ,\phi )=R_{n\ell }(r)Y_{\ell m}(\theta ,\phi )$$ubi $Y_{\ell m}(\theta ,\phi )$ sunt functiones (vel harmonicae) sphericae et $R_{n\ell }(r)$ partes radiales vicissim productō functiones exponentiales ac polynomii Laguerrei $L_{n-\ell -1}^{2\ell +1}$ exprimantur: $$R_{n\ell }(r)=\sqrt{{\left(\frac{2Z}{n}\right)}^3\frac{(n-\ell -1)!}{2n(n+\ell )!}}{\left(\frac{2Zr}{n}\right)}^{\ell }{L}_{n-\ell -1}^{2\ell +1}\left(\frac{2Zr}{n}\right)\exp (-\frac{Zr}{n})$$

Numerus $n-\ell -1$ est enim quot adsunt nodi vel zeri vel radicis functionis radialis.

Energia $E$ functionis propriae cum numeribus quanticis $n\ell m$ solō a numerō principale $n$ pendet:$$E_{n\ell m}=-\frac{1}{2}\frac{Z^2}{n^2}$$Haec est manifestatio "degenerationis inexpectatae" quae de symmetriā atomi hydrogenii respectu catervae $SO(4)$ - illae rotationum quadridimenionalium et non solō tridimensionalium oritur.

Hae sunt solutiones problematis atomi hydrogeniformae (hydrogenoidi) id est atomi (ionis) cum unō electrone in campo nuclei oneris $+Z\mathrm{e}$.

Structura molecularum/molecularis

Principium exclusionis Pauli

Index magnitudinum physicarum

Ŏpŭs latinum de chemia quantica tractans

"Nova methodus adhibendi approximationem molecularium orbitalium ad plures iuxtapositas unitates" devulgatum a M. Suard, G. Berthier et G. Del Re in Theoretica Chemica Acta 7 (3): 236–244 in lingua Anglica traductum cum commentariis

Bibliographia

• Г. Гельман (G. Gel’man; Moscow - 23 Oct 1936). Квантовая Химия [Quantum Chemistry (in Russian)]. ОНТИ, Москва и Ленинград [ONTI, Moscow and Leningrad] (1937), 546 с. [C. 1937 II, 3284] Translated from the German manuscript to Russian by J. Golovin, N. Tunitskij, M. Kovner. Volume 1 of the series “Physics in Monographs”, edited by S. I. Vavilov, I. Ye. Tamm and E. V. Shpolskii.

• H. Hellmann (Moscow - Mar 1937). Einführung in die Quantenchemie [Introduction to Quantum Chemistry (in German)]. Deuticke, Leipzig and Wien (1937), VIII + 350 p. [C. A. 31:77371, CAN 31:55845 / C. 1937 II, 1518]. Book review: J. Syrkin, Acta Physicochim. U.R.S.S. 8 (1938) 138-140 Book review: O. Schmidt, Z. Elektrochem. Angew. Phys. Chem. 44 (1938) 284 (DOI: 10.1002/bbpc.19380440415) Book review: Clusius, Angew. Chem. 54 (1941) 156 (DOI: 10.1002/ange.19410541109) Reprint (reproduction vested by Alien Property Custodian): H. Hellmann. Einführung in die Quantenchemie. J. W. Edwards, Ann Arbor, Michigan (1944), VIII + 350 p. [C. A. 38:54585, CAN 38:36665]

Nexus externi

• Situs interretialis ab Udo Anders sustinetur "Some details, hopefully entertaining, in and around early quantum chemistry (1927—1980)"