Usor:Tchougreeff/Orbitalia hybridica (hybridae)

Orbitalia hybridica (hybridae) sunt combinationes lineares orbitalium atomicōrum ad solam unam atomam pertinentes. Ut quisque status mechanicae quanticae orbitalia hybridica (vel hybridae) illius vel alius experimentum describendi servant. Specialiter, orbitalia molecularia conveniunt ad experimenta describendum quae totam moleculam probant (?). In contrario hybridae testibus localibus et supra omnibus chemicis responderunt. Exempli gratia in atomo carbonii quattuor ligamina formante 2s orbital cum tribus 2p orbitalibus quattuor aequivalentes sp³ mixturas formandi causā combinationes lineares cum coefficientibus $\pm \!1/\!2$ ubi signa selecta sint ... in a tetrahedral arrangement around the carbon to bond to four different atoms. Hybrid orbitals are useful in the explanation of molecular geometry and atomic bonding properties and are symmetrically disposed in space.

Saepissime combinationes orbitalium s et p considerantur. Et inter eis saepissime orbitalia hybridica formarum fixarum sp³ vel sp² considerantur.

Nos autem proposuerāmus orbitalia hybridica formarum flexibilĭum ūti. Orbital hybridicum flexibile est casus particularis combinationis linearis orbitalium atomicorum (purorum):

$$\left|t_{Am}\right\rangle =\sum _{a\in A}\left|a\right\rangle \left\langle a\mid m\right\rangle$$

ubi $a$ est orbital atomicum (purum, id est $a=s,p_{x},p_{y},p_{z},$) in atomo A; $\left|t_{Am}\right\rangle$ est orbital hybridicum pertinens ad atomam A et ligamen m ($t=r,l$), ejusque expansionis coefficientes $\left\langle a\mid m\right\rangle$ sunt. Haec formula utens notationem nōnnihil complicatam exprimit combinationem linearem orbitalium atomicorum ubi coefficientes pārent orthonormalitatis conditioni propter quod eos matricem gregi $SO(4)$ pertinentem formant. Tametsi quaelibet antedictae matrices 16 elementa habent propter othonormalitatem solo 6 variabiles independentes (angulorum Jacobii) sates sunt ad eam definiendum:

$$h=J(\omega _{sx})J(\omega _{sy})J(\omega _{sz})J(\omega _{yz})J(\omega _{xz})J(\omega _{xy})$$

ubi $J(\omega _{ij})$ matrices rotationum Jacobii sunt:

$$J(\omega _{ij})={\begin{pmatrix}1&...&...&...&0\\\\...&\cos \omega _{ij}&...&\sin \omega _{ij}&...\\&&1&&\\...&-\sin \omega _{ij}&...&\cos \omega _{ij}&...\\&&&&\\0&...&...&...&1\\\end{pmatrix}}$$

Anguli $\omega _{sx},\omega _{sy},\omega _{sz},\omega _{yz},\omega _{xz},\omega _{xy}$ vicissem duos vectores ${\vec {\omega }}_{b}=(\omega _{sx},\omega _{sy},\omega _{sz}),{\vec {\omega }}_{l}=(\omega _{yz},\omega _{xz},\omega _{xy})$ formant, qui mutuo formam hybridarum ac earum orientationem describunt.

Alio modo eadem rotatio duobus quaternionibus normalizatis representari potest. Enimvero, ipsae hybridae notandae sunt quaternionibus representabiles esse, quia quattuor coefficientes orbitalium atomicorum in partem scalarem - coefficiens orbitalis s - et partem vectorialem - tres coefficientes apud orbitalia p - dīvĭduntur. In hāc picturā relatio duārum hybridārum ${\textstyle h^{\prime },h}$ pulcherrime:

$$h^{\prime }=p\circ h\circ q,$$

conjuguntur, ubi ${\textstyle \circ }$ multiplicationem quaternionum significat et quaterniones normalizatae p et q antedictis vectoribus ${\vec {\omega }}_{b},{\vec {\omega }}_{l}$ dēfīniuntur:

$${\begin{array}{c}q_{0}=\cos {\frac {\omega _{+}}{2}},q_{x}={\frac {\omega _{+x}}{\omega _{+}}}\sin {\frac {\omega _{+}}{2}},q_{y}={\frac {\omega _{+y}}{\omega _{+}}}\sin {\frac {\omega _{+}}{2}},q_{z}={\frac {\omega _{+z}}{\omega _{+}}}\sin {\frac {\omega _{+}}{2}},\\p_{0}=\cos {\frac {\omega _{-}}{2}},p_{x}={\frac {\omega _{-x}}{\omega _{-}}}\sin {\frac {\omega _{-}}{2}},p_{y}={\frac {\omega _{-y}}{\omega _{-}}}\sin {\frac {\omega _{-}}{2}},p_{z}={\frac {\omega _{-z}}{\omega _{-}}}\sin {\frac {\omega _{-}}{2}},\end{array}}$$

ubi est

$$\omega _{\gamma \pm }=\epsilon _{\alpha \beta \gamma }\omega _{\alpha \beta }\pm \omega _{s\gamma },$$

et

$$\omega _{\pm }={\sqrt {\sum _{\gamma }\omega _{\pm \gamma }^{2}}},\gamma =x,y,z$$