|  |  |
| --- | --- |
|  | **Haec commentatio vicificanda est ut rationibus qualitatis propositis obtemperet.**Quapropter rogamus ut corrigas, praecipue introductionem, formam, nexusque extra et intra Vicipaediam. |

**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 {\textstyle \psi (\mathbf {x} \_{1},\mathbf {x} \_{2},\ldots ,\mathbf {x} \_{N},t)} exprĭmĭtur. Hujus functionis argumenta (variabiles) sunt omnium systematis corpōrum coordinatae {\textstyle \mathbf {x} \_{1},\mathbf {x} \_{2},\ldots ,\mathbf {x} \_{N}} et tempus {\textstyle t}. Functiones undariae legi Schrödingeri: 


i
ℏ

∂

∂
t

ψ
=
H
ψ

{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =H\psi }

{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =H\psi }subordinantur, ubi operator Hamiltonianus {\displaystyle H} (in chemia quantica) a tempore non pendet. Ponatur {\textstyle \psi (\ldots ,t)=\psi (\ldots )\exp(-{\dfrac {iEt}{\hbar }})}, ubi {\displaystyle E} quidam numerus realis est, pars prima (laeva) suprae aequationis Schrödingeri erit {\displaystyle E\psi (\ldots )\exp(-{\frac {iEt}{\hbar }})}, et ambae aequationis partes, multiplicatore exponentiale divisis, dabunt 


H
ψ
(
…
)
=
E
ψ
(
…
)

{\displaystyle H\psi (\ldots )=E\psi (\ldots )}

{\displaystyle H\psi (\ldots )=E\psi (\ldots )}
ubi tempus ({\displaystyle 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 {\displaystyle H} (vide infra) ergo per functionem undariam dīvĭdi non posset. Aequatio statica multam solutionem habet quaevis quārum quemdam status quanticum systematis Hamiltoniano {\displaystyle H} descriptis praebet. Functio autem undaria solō a variabilibus {\textstyle \mathbf {x} \_{1},\mathbf {x} \_{2},\ldots ,\mathbf {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

n
n

+

V

n
e

+

V

e
e

{\displaystyle H=T\_{n}+T\_{e}+V\_{nn}+V\_{ne}+V\_{ee}}

{\displaystyle H=T_{n}+T_{e}+V_{nn}+V_{ne}+V_{ee}}ubi {\textstyle T\_{n}} energia kinetica nuleōrum (suum operatrum): 



T

n

=
−

ℏ

2

2

∑

α
=
1

A

1

M

α


∇

α

2

{\displaystyle T\_{n}=-{\frac {\hbar ^{2}}{2}}\sum \limits \_{\alpha =1}^{A}{\frac {1}{M\_{\alpha }}}\nabla \_{\alpha }^{2}}

{\displaystyle T_{n}=-{\frac {\hbar ^{2}}{2}}\sum \limits _{\alpha =1}^{A}{\frac {1}{M_{\alpha }}}\nabla _{\alpha }^{2}}est (hic {\displaystyle A} numerus nucleōrum est, {\displaystyle M\_{\alpha }} massa {\displaystyle \alpha }-isimi nuclei est); similiter {\textstyle T\_{e}} energia kinetica electronum: 



T

e

=
−

ℏ

2

2

m

e



∑

i
=
1

N

∇

i

2

{\displaystyle T\_{e}=-{\frac {\hbar ^{2}}{2m\_{e}}}\sum \limits \_{i=1}^{N}\nabla \_{i}^{2}}

{\displaystyle T_{e}=-{\frac {\hbar ^{2}}{2m_{e}}}\sum \limits _{i=1}^{N}\nabla _{i}^{2}}est ({\displaystyle m\_{e}} est massa electronis). Enim operatora {\displaystyle V\_{nn},V\_{ne},V\_{ee}} sunt quodque energiae potentialis interactionis electrostaticae inter nucleis 



V

n
n

=


e

2

2

∑

α
≠
β


Z

α

Z

β


|

R

α

−

R

β

|


{\displaystyle V\_{nn}={\dfrac {\rm {e^{2}}}{2}}\sum \limits \_{\alpha \neq \beta }{\dfrac {Z\_{\alpha }Z\_{\beta }}{|{\bf {R}}\_{\alpha }-{\bf {R}}\_{\beta }|}}}

{\displaystyle V_{nn}={\dfrac {\rm {e^{2}}}{2}}\sum \limits _{\alpha \neq \beta }{\dfrac {Z_{\alpha }Z_{\beta }}{|{\bf {R}}_{\alpha }-{\bf {R}}_{\beta }|}}} (hic {\displaystyle Z\_{\alpha }} onus electricus {\displaystyle \alpha }-isimi nuclei in unitatibus oneris electronis {\displaystyle {\rm {e}}} est - ergo aliqui numerus naturalis minor quam 118, {\displaystyle {\bf {R}}\_{\alpha }} vector tri-dimensionalis loci {\displaystyle \alpha }-isimi nuclei est); energiae potentialis ineractionis electrostaticae inter nucleis et electronibus: 



V

n
e

=
−

e

2


∑

α
j



Z

α

|

R

α

−

r

j

|


{\displaystyle V\_{ne}=-{\rm {e^{2}}}\sum \limits \_{\alpha j}{\dfrac {Z\_{\alpha }}{|{\bf {R}}\_{\alpha }-{\bf {r}}\_{j}|}}}

{\displaystyle V_{ne}=-{\rm {e^{2}}}\sum \limits _{\alpha j}{\dfrac {Z_{\alpha }}{|{\bf {R}}_{\alpha }-{\bf {r}}_{j}|}}}(hic {\displaystyle {\bf {r}}\_{j}} vector loci {\displaystyle j}-simi electronis est); energiae potentialis interactionis electronum: 



V

e
e

=


e

2

2

∑

i
≠
j


1

|

r

i

−

r

j

|


.

{\displaystyle V\_{ee}={\dfrac {\rm {e^{2}}}{2}}\sum \limits \_{i\neq j}{\dfrac {1}{|{\bf {r}}\_{i}-{\bf {r}}\_{j}|}}.}

{\displaystyle V_{ee}={\dfrac {\rm {e^{2}}}{2}}\sum \limits _{i\neq j}{\dfrac {1}{|{\bf {r}}_{i}-{\bf {r}}_{j}|}}.}Notandum interdum est, quod variabiles ad electrona pertinentes {\displaystyle {\bf {x}}\_{j}}, a quibus functio undaria pendet, non solō tri-dimensionales locuum vectores {\displaystyle {\bf {r}}\_{j}=(r\_{xj},r\_{yj},r\_{zj})} sunt, sed autem variabiles electronum spirulares {\displaystyle s\_{j}}, quae unum ambōrum valōrum ±½ admissibilium pro fermione spirulitatis vel volubilitatis ½ accipit: {\displaystyle {\bf {x}}\_{j}=({\bf {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 {\displaystyle T\_{n}} magno minor est quam energia kinetica electronum {\displaystyle T\_{e}} ita ut illa neglecta esse possit. Ita familiam Hamiltonianōrum electronicōrum habemus 



H

e

=

T

e

+

V

n
n

+

V

n
e

+

V

e
e

{\displaystyle H\_{e}=T\_{e}+V\_{nn}+V\_{ne}+V\_{ee}}

{\displaystyle H_{e}=T_{e}+V_{nn}+V_{ne}+V_{ee}}cujus membra copiā locōrum nucleōrum {\textstyle \{{\bf {R}}\_{\alpha }|\alpha =1,\ldots ,A\}} distinguuntur. Physice, autem, quaeque data configuratio nucleōrum {\textstyle \{{\bf {R}}\_{\alpha }|\alpha =1,\ldots ,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 {\textstyle \Psi \_{e}(\mathbf {x} \_{1},\mathbf {x} \_{2},\ldots ,\mathbf {x} \_{N})}, eandem functio undaria electronica nōmĭnēmus, exprĭmuntur. Quoniam Hamiltonianum electronicum pro quōque configuratione nucleōrum {\textstyle \{{\bf {R}}\_{\alpha }\}} potentialem electricam congruentem habet, functio undaria electronica aequationis Schödingeri staticae sătisfăciens aeque ab configuratione {\textstyle \{{\bf {R}}\_{\alpha }\}} pendet; ergo scribere possumus {\displaystyle \Psi \_{e}=\Psi \_{e}(\{\mathbf {x} \_{j}\}\mid \{{\bf {R}}\_{\alpha }\})} ad dependentiam functionis electronicae ab configuratione {\displaystyle \{{\bf {R}}\_{\alpha }\}} denotandum. Enimvero energia tota electronica quid est valor medius exspectatus Hamiltoniani electronici {\displaystyle E\_{e}=\langle \Psi \_{e}|H\_{e}|\Psi \_{e}\rangle \_{\mathbf {x} }} est autem finctio configurationis {\displaystyle E\_{e}=E\_{e}(\{{\bf {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 {\displaystyle Z} (numerō positivō integrō) compendii gratiā scribere possumus:



H
^

ψ
(

r

)
=

(

−

ℏ

2

∇

2

2

m

e



−

Z

e

2

|

r

|


)

ψ
(

r

)
=
E
ψ
(

r

)

{\displaystyle {\hat {H}}\psi (\mathbf {r} )={\bigl (}-{\frac {\hbar ^{2}\nabla ^{2}}{2m\_{\mathrm {e} }}}-{\frac {Z{\mathrm {e} }^{2}}{\left\vert \mathbf {r} \right\vert }}{\bigr )}\psi (\mathbf {r} )=E\psi (\mathbf {r} )}

{\displaystyle {\hat {H}}\psi (\mathbf {r} )={\bigl (}-{\frac {\hbar ^{2}\nabla ^{2}}{2m_{\mathrm {e} }}}-{\frac {Z{\mathrm {e} }^{2}}{\left\vert \mathbf {r} \right\vert }}{\bigr )}\psi (\mathbf {r} )=E\psi (\mathbf {r} )}

quid in unitatibus atomicis transcribi potest 



H
^

ψ
(

r

)
=

(

−

∇

2

2

−

Z

|

r

|


)

ψ
(

r

)
=
E
ψ
(

r

)
.

{\displaystyle {\hat {H}}\psi (\mathbf {r} )={\bigl (}-{\frac {\nabla ^{2}}{2}}-{\frac {Z}{\left\vert \mathbf {r} \right\vert }}{\bigr )}\psi (\mathbf {r} )=E\psi (\mathbf {r} ).}

{\displaystyle {\hat {H}}\psi (\mathbf {r} )={\bigl (}-{\frac {\nabla ^{2}}{2}}-{\frac {Z}{\left\vert \mathbf {r} \right\vert }}{\bigr )}\psi (\mathbf {r} )=E\psi (\mathbf {r} ).}Ut eam resolvāmus a coordinatis cartesianis ad coordinatas sphaericas 



r

=
(
x
,
y
,
z
)
=
r
(
sin
⁡
θ
cos
⁡
φ
,
sin
⁡
θ
sin
⁡
φ
,
cos
⁡
θ
)

{\displaystyle \mathbf {r} =(x,y,z)=r(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )}

{\displaystyle \mathbf {r} =(x,y,z)=r(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )}transeāmus, in quibus operator Laplacianus in coordinatis Cartesianis formam: 



∇

2

=

∂

2

∂

x

2


+

∂

2

∂

y

2


+

∂

2

∂

z

2


{\displaystyle \nabla ^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}

{\displaystyle \nabla ^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}} habens, accipit autem formam



∇

2

=

1

r

2



∂

∂
r

(

r

2

∂

∂
r

)

+

1

r

2

sin
⁡
θ


∂

∂
θ

(

sin
⁡
θ

∂

∂
θ

)

+

1

r

2

sin

2

⁡
θ



∂

2

∂

φ

2


.

{\displaystyle \nabla ^{2}={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2} \over \partial \varphi ^{2}}.}

{\displaystyle \nabla ^{2}={1 \over r^{2}}{\partial  \over \partial r}\left(r^{2}{\partial  \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial  \over \partial \theta }\left(\sin \theta {\partial  \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2} \over \partial \varphi ^{2}}.}Quae autem simplicius videtur quia cum debita dexteritate adhibenda functio undaria electronica in campō nucleō onere {\displaystyle Z} inductō in productō partium duārum/factōrum duōrum dissolvi potest: 



ψ

n
ℓ
m

(
r
,
θ
,
φ
)
=

R

n
ℓ

(
r
)

Y

ℓ
m

(
θ
,
φ
)

{\displaystyle \psi \_{n\ell m}(r,\theta ,\varphi )=R\_{n\ell }(r)Y\_{\ell m}(\theta ,\varphi )}

{\displaystyle \psi _{n\ell m}(r,\theta ,\varphi )=R_{n\ell }(r)Y_{\ell m}(\theta ,\varphi )}ubi {\textstyle Y\_{\ell m}(\theta ,\varphi )} sunt functiones (vel harmonicae) sphericae et {\textstyle R\_{n\ell }(r)} partes radiales vicissim productō functiones exponentiales ac polynomii Laguerrei {\textstyle L\_{n-\ell -1}^{2\ell +1}} exprimantur: 



R

n
ℓ

(
r
)
=


(

2
Z

n

)

3


(
n
−
ℓ
−
1
)
!

2
n
(
n
+
ℓ
)
!


(

2
Z
r

n

)

ℓ

L

n
−
ℓ
−
1

2
ℓ
+
1

(

2
Z
r

n

)

exp
⁡

(

−

Z
r

n

)

{\displaystyle 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 {\left(-{\frac {Zr}{n}}\right)}}

{\displaystyle 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 {\left(-{\frac {Zr}{n}}\right)}}
Numerus {\displaystyle n-\ell -1} est enim quot adsunt nodi vel zeri vel radicis functionis radialis.

Energia {\displaystyle E} functionis propriae cum numeribus quanticis {\displaystyle n\ell m} solō a numerō principale {\displaystyle n} pendet:



E

n
ℓ
m

=
−

1
2


Z

2

n

2



{\displaystyle E\_{n\ell m}=-{\frac {1}{2}}{\frac {Z^{2}}{n^{2}}}}

{\displaystyle E_{n\ell m}=-{\frac {1}{2}}{\frac {Z^{2}}{n^{2}}}}Haec est manifestatio "degenerationis inexpectatae" quae de symmetriā atomi hydrogenii respectu catervae {\displaystyle 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 {\displaystyle +Z\mathrm {e} }. 

#### Atomi cum pluribus electronibus

### 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)"

1. ↑
2. ↑  Hic subscriptum {\displaystyle {\mathrm {x} }} significat integrationem solo respectu variabilium electronicarum valorem medium calculando.