Usor:Tchougreeff/QUOMODO sive HOW TO/SBDR

Nova Methodus Adhibendi Approximationem Molecularium Orbitalium ad Plures Iuxtapositas Unitates

MICHÈLE SUARD et GASTON BERTHIER Laboratoire de Chimie Quantique de la Faculté des Sciences, Paris GIUEPPE DEL RE Gruppo Chimica Teorica del CNR, Rome Acceptum a.d. XVIII Kal. Ianuarias a. MCMLXVII p. chr. n. Theoret. chim. Acta (Berl.) 7, 236-244 (1967). Abstract The study of the electronic structure of a chain formed by repetition of polyatomic monomers and the interpretation of the resulting polymer are made much easier if the interaction terms relating to neighbouring units can be reduced to a single parameter. This problem has been treated in the MO scheme by a procedure which makes it possible to take into account the modifications of the monomer within the polymer and only neglects minor contributions. The interaction bet ween two monomers i and j is represented by the n x n submatrix B_{ij} formed by the corresponding non diagonal terms of the energy matrix. By appropriate unitary transformations it is possible to bring the submatrices B_{ij} of neighbouring monomers to a diagonal form. In general, one of the diagonal elements thus obtained is much higher than the others, and can be taken as a measure of the i-j interaction. A numerical application has been made in the case of a fictitious polymer formed by H_{2} molecules, and in the case of a polypeptide formed by HNCO groups interacting through hydrogen bonds. The validity of the approximations introduced for different values of interaction can be tested on a dimer whose units are placed at different distances. In the case of a polymer the transformations mentioned above require that the atomic orbitals of a monomer be replaced by orbitals extending over two neighbouring units. Taking as a new unit the corresponding pair of monomers, the interaction between the units can be represented by a single term. The reduction of the B_{ij} matrices to a single term each gives energy bands whose positions are in perfect agreement with those obtained directly by the classical techniques of solid-state physics [7]. Abstract Des Studium der elektronischen Struktur von linearen Polymeren vereinfacht sich sehr, wenn die Wechselwirkung zwischen benachbarten Monomeren durch einen Parameter beschrieben werden karm. Entsprechend wird im MO-Schema ein Verfahren entwiekelt, das auf der Veränderung der Monomeren im Polymeren fußt. Die Wechselwirkung zwischen zwei Monomeren i,j des Polymer wird durch eine aus Nichtdiagonalelementen der Hamiltonmatrix bestehende Untermatrix B_{ij} beschrieben. Für alle benachbarten Monomerenpaare lassen sich diese durch unitare Transformationen diagonalisieren. I. a. ist eines der so erhaltenen Diagonalelemente von B merklich größer und dient als Maß für die Wechselwirkung der beiden Monomeren. Angewendet wird die Methode auf ein fiktives Polymer aus H_{2}-Molekülen und auf ein Polymer aus durch H-Brücken verbundenen Peptidgruppen. Der Einfluß der Näherungen wird zunaehst an einem Dimeren für verschiedene Abstände getestet. Im Falle eines Polymeren sind als Monomere Einheiten von zwei benachbarten Molekülen mit entsprechend ausgedehnten Orbitalen zu betrachten. Die Rechnung führt hier auf Energiebänder, deren Lage voll mit der nach klassischen Methoden der Festkörperphysik [7] erhaltenen übereinstimmt. Résumé L'étude de la structure électronique d'une chaîne constituée par la répétition d'un motif polyatomique et l'interprétation des propriétés du polymère ainsi formé sont grandement facilités, s'il est possible de ramener les termes d'interaction entre unités voisines d'un paramètre d'interaction unique. Ce problème a été traité en méthode des orbitales moléculaires par un procédure qui permet de tenir compte des modifications du monomère à l'intérieur du polymère et ne néglige que de faibles contributions. L'interaction entre deux monomères i et j est représentée par la sous-matrice B_{ij} de dimension n • n formée par les termes non-diagonaux correspondants de la matrice-énergie. Par des transformations unitaires appropriées, il est possible de mettre sous forme diagonale les sousmatrices B_{ij} entre monomères voisins. En général, la diagonale de la matrice transformée contient un terme prépondérant qu'on peut considérer comme mesurant l'interaction (i, j). Une application numérique a été effectuée dans le cas d'un polymère fictif formé par des molécules H2 et dans le cas d'un polymère peptidique formé par des groupements HNCO en interaction par l'intermédiaire de liaisons hydrogène. La validité des approximations introduites selon l'importance de l'interaction peut être vérifiée sur l'exemple d'un dimère dont les deux unités sont placées s des distances variables. Dans le cas d'un polymère les transformations effectuées conduisent ~ remplacer les orbitales atomiques du monomère par des orbitales s'étendant sur deux unités voisines; en prenant comme motif structural de la chaîne le monomère double formé par deux unités successives, on peut représenter l'interaction entre deux motifs adjacents par un seul terme. La réduction de la matrice B_{ij} à un seul terme fournit des bandes d'énergie dont la position est en très bon accord avec celles obtenues directement par les techniques classiques de calcul de l'état solide [7].
Cum conductionem transmissionemque electronum in moleculis polypeptidum consideravissemus [10], id nobis dignum studio visum est, statum uniuscuiusque partis seu monomeris [6] totius moleculae ita describere ut, quantum fieri posset, interactio monomeris cum monomere singula mensura repraesentaretur. Cuius problematis duo sunt proposita: primum, ut status monomeris in polymere simpliciore modo describatur; alterum, ut quaedam tantum parvae contributiones negligantur, cum proprietates longioris polymeris per extrapolationem deducere velimus. Utrumque propositum est persequendum, etiamsi longissimum polymeres methodo Egr. Vir. KOUTECKY [7] possit tractari directe, id quod inter alios Egr. Vir. LADIK [8] ostendit; habet enim extrapolatio virtutem quandam e simplicibus ad complicata procedendi, itaque, dummodo caute adhibeatur, simul ad bonam descriptionem et ad claram interpretationem phenomenorum ducit.	Since we <previously> had considered conduction and transmission of electrons in polypeptide molecules [10], it seemed to us to be worth so to describe state of each part, or monomer [6], of the whole molecule that, as much as it would be possible, the interaction of <one> monomer with <another> monomer would be represented by a single measure. Objectives of this setting are twofold: one, to describe the state of a monomer in a polymer in a simpler way; another, to neglect some so small contributions that properties of longer polymers would be deduced by an extrapolation. Whatever of these objectives being pursued (despite even longest polymers can be trated directly by the KOUTECKY method [7] as LADIK [8] have shown among others) the extrapolation has an advantage/value of enabling a progress from simpler to more involved as well as if applied cautiously, leads simultaneously to a good description and clear interpretation of phenomena.
1 Propositiones Generales Quomodo status monomeris et polymeris sit describendus theoretice, ante omnia statuamus. Descriptionem status monomeris et polymeris per linearem combinationem atomicorum orbitalium (LCAO) obtentam adhibebimus; paulum hic nostra interest, quomodo coefficientes linearis combinationis obtenti sint, quia non valores elementorum matricum in quavis data basi magni sunt momenti, sed eorum variatio atque commixtio, quae tum fiunt, cum vel basis vel ligamina variantur. Iuxtaposita fingamus nunc aliquot monomera noninteragentia, quasi in diversis universis posita. Matrix $A$ totius ficti polymeris ita obtenti erit summa directa ($\bigoplus$) minorum matricum (quae vocentur $A_{1},A_{2}$, etc.) alia cum alio monomere ab aliis seiuncto correspondente. Ligamina inter monomera hoc efficiunt, ut mutentur $A_{1},A_{2}$, etc., et in $A$ aliquot nova elementa extra matrices $A_{i}$ appareant:	General propositions Let us establish first of all, how the state of a monomer of polymer is to be described theoretically. We shall apply a description of the state of a monomer of polymer obtained by the linear combination of atomic orbitals (LCAO); it is of minor interest for us, how the coefficients of the linear combinations have been obtained, since not the <specific> values of the matrix elements in whatever basis are important, rather their variation and mixing, which appear, when either basis or bonds vary. Now, let us imagine some number of noninteracting monomers, as if they had existed in different universes. Matrix $A$ of so obtained whole fictitious polymer will be a direct sum ($\bigoplus$) of smaller matrices (which are called $A_{1},A_{2}$, etc.) each corresponding to respective monomer detached from others. The bonds between monomers act so that $A_{1},A_{2}$, etc. vary and in $A$ some new elements appear outside of matrices $A_{i}$:
$A=\oplus A_{i}\rightarrow A^{\prime }=A+\oplus \Delta A_{i}+R,(1)$
ubi	where
$R=\left|{\begin{array}{cccc}0&B_{12}&B_{13}&\ldots \\B_{12}^{\dagger }&0&B_{23}&\ldots \\\ldots &\ldots &\ldots &\ldots \\\ldots &\ldots &\ldots &\ldots \end{array}}\right|.$
Quemadmodum illi solent, qui moleculas Hückelii methodo tractant, matrices $B_{ij}$ interactionem inter monomera describere assumimus; illae autem, quibus nomen $\Delta A_{i}$ dedimus, primum, adaptationem status monomeris ab aliis seiuncti novae spatii structurae ac conformationi, ut ita dicam, significant; secundum, "promotiones" quasdam electronum a proximitate aliorum monomerium inductas, e.g. mutationes earum energiarum, quae ad extrahendos electrones ex monomere ab aliis seiuncto opus sunt, quodammodo numerice exprimunt. Breviter, secundum Cl. Vir. MULLIKEN [9], $A_{i}+\Delta A_{i}$ matrix hamiltoniana monomeris "in situ" potest vocari. Definitio autem monomeris in situ ex ante dictis clare apparet; quaestio, num revera monomeres in situ univoce cum monomere ab aliis seiuncto correspondeat, hic non considerabitur, etsi maximi momenti est, quia nostra nunc interest specialiter de $B_{ij}$ matricibus tractare. In hac enim dissertatiuncula problematis supra positi tantum id, quod ad interactionem monomeris cum monomere refertur, examinabitur, aliis rebus ad alias dissertationes remissis. Unam ex multiplicitate basium aliarum aliis per unitarias transformationes aequivalentium eligere non oportet, dum aut $A$ aut $A+\oplus \Delta A_{i}$ considerentur. Cum tamen usui $B_{ij}$ attendimus, vel ad res melius describendas et interpretandas, vel ad simpliciores calculos efficiendos, electioni basis maxima cura est adhibenda. Cuius rei causa haec est, matrices $B_{ij}$ simplicissimas esse, id est ex uno elemento consistere, si simplicissima systemata (e.g., π electrones coniugatae moleculae) tractentur; ex multis contra elementis consistere, cum monomera non sint atomi unum orbitale parantes, sed complicatae moleculae; unde illic facile est $B_{ij}$ tamquam mensuram interactionis interpretari, hic difficillimum. Aliquid tamen, nostra quidem sententia, facere possumus, ut saepe matrix interactionis inter vicina monomera $B_{ij}$ infra semper ad interactionem inter vicina monomera referetur; interactiones autem inter non vicina monomera neglegentur., etiamsi non sit primi ordinis, ad unum elementum reducatur. Nam, ut alio loco dictum est [3], semper per reales transformationes unitarias quaelibet realis matrix $B_{ij}$ ad diagonalem $b_{ij}$ reduci potest:	In the way customary to those who treat molecules by the Hückel method, we assume matrices $B_{ij}$ to describe interaction between monomers; those, meanwhile, to which we gave the name $\Delta A_{i}$, first, so to say, signify adaptations of the states of the detached monomers to the new spatial structure or conformation; second, they express "????" of some electrons induced by proximity of other monomers, e.g. variations of their energies necessary for extracting electrons from monomers detached from others, whatever are their numerical values. Shortly, according to MULLIKEN [9], $A_{i}+\Delta A_{i}$ can be called Hamiltonian matrix of a monomer "in situ". Meanwhile the definition of a monomer in situ is clearly seen from the said above; whereas the question whether the in situ monomers indeed one-to one correspond to a monomer detached from others will not be considered here, although it is of great importance, since our interest here is specifically treat matrices $B_{ij}$. Thus in the present paper only those problems which refer to the interaction of a monomer with monomer will be examined, other issues being left to other papers. It not ought to select one of multiple equivalent basis <sets> <connected> by unitary transformations while either $A$ or $A+\oplus \Delta A_{i}$ would be considered. When however we turn toward the usage of $B_{ij}$ either for better describing and interpreting things or for making calculations simpler, the great care must be applied to the selection of the <suitbale> basis. The reason to do so is to make matrices $B_{ij}$ as much simple as possible, that is to consist of one element <at least if> simplest systems (e.g. π electrons of a coniugate molecule) are treated, by contrast to consist from numerous elements, provided monomers are not atoms bearing one orbital, but complex molecues, so that those $B_{ij}$ are easily interpreted as measure of interactions, these more difficult. ... On the other hand, as it is said in another place [3], whatevr real matrix $B_{ij}$ can be always reduced to the diagonal $b_{ij}$:
$V_{i}^{\dagger }B_{ij}V_{j}=b_{ij}(II)$
dummodo $V_{i}$matricem ($B_{ij}B_{ji}$), $V_{j}$ matricem ($B_{ji}B_{ij}$) ad diagonalem reducat:	so that $V_{i}$ reduces to diagonal matrix ($B_{ij}B_{ji}$), <and> $V_{j}$ matrix ($B_{ji}B_{ij}$)
$(B_{ij}B_{ji})V_{i}=V_{i}(b_{ij})^{2},(B_{ji}B_{ij})V_{j}=V_{j}(b_{ij})^{2}.(IIbis)$
Itaque possumus assumere matricem $B_{ij}$, cuius ordo generaliter $n$ est (si $n$ numerum significat orbitalium monomeris), non $n^{2}$ sed $n$ tantum elementa habere, transformationibus $V_{i}$ atque $V_{j}$ adhibitis. In plerisque casibus, unum tantum elementum matricis diagonalis $b_{ij}$ a nullo multum differt: quod elementum tamquam mensuram interactionis vere, quamvis approximate, assumere possumus. Matrix $V$, quae summa directa matricum $\oplus V_{i}$ est,	This way we can assume matrix $B_{ij}$, whose order is in general $n$ (if $n$ denotes the number of orbitals in a monomer) to have not $n^{2}$ but $n$ elements, provided transformations $V_{i}$ and $V_{j}$ are applied. In the majority of cases only one element of the diagonal matrix $b_{ij}$ significantly differs from zero: we can assume this element as true, although approximate, measure of interaction. Matrix $V$, which is a direct sum of matrices $\oplus V_{i}$ :
$V=\oplus V_{i},(III)$
transformationem unitariam ejus basis, ad quam $A$ initio est relata, definit; quid exacte $V$ efficiat, ad matricem $A$ adhibita, est nunc considerandum.	defines a transformation of the basis in which $A$ is written initially; now it is to consider how exactly the matrix $V$ would act when applied to matrix $A$.
2 Casus Duorum Monomerium Dimeres ante consideremus. Matrix $A^{\prime }$ sit igitur:	Case of two monomers First of all let us consider dimers. Therefore, matrix $A^{\prime }$ would be:
$A^{\prime }=\left|{\begin{array}{cc}A_{1}+\Delta A_{1}&B_{12}\\B_{12}^{\dagger }&A_{2}+\Delta A_{2}\end{array}}\right|(IV)$
Transformatio autem $V$ erit:	Transformation $V$ will be
$V=\left|{\begin{array}{cc}V_{1}&0\\0&V_{2}\end{array}}\right|(v)$
unde	where from
${\tilde {A}}^{\prime }=V^{\dagger }A^{\prime }V=\left|{\begin{array}{cc}V_{1}^{\dagger }\left(A_{1}+\Delta A_{1}\right)V_{1}&V_{1}^{\dagger }B_{12}V_{2}\\V_{2}^{\dagger }B_{12}^{\dagger }V_{1}&V_{2}^{\dagger }\left(A_{2}+\Delta A_{2}\right)V_{2}\end{array}}\right|(VI)$
Ut rem definiamus, duas hydrogeni moleculas, ita iuxtapositas ut nuclei in eadem recta jaceant, fingamus:	For the sake of definiteness let us imagine two hydrogen molecues so mutually positioned that the nuclei would lay on the same strait <line>.
$\underbrace {{\overset {(1)}{H}}-{\overset {(2)}{H}}} _{a}\underbrace {.................} _{b}\underbrace {{\overset {(3)}{H}}-{\overset {(4)}{H}}} _{a}.(VII)$
Referentiam energiarum unitatemque ita eligamus, ut diagonalia elementa matricis $A^{\prime }$ nulla sint, atque ut constans $k$ sequentis formulae [4] sit unitas:	Let us select the reference <point of> energy so that diagonal elements of the matrix $A^{\prime }$ would be zero, as so that the constant $k$ of the following equation [4] would be unity:
$A_{ij}^{\prime }=-k{\frac {S_{ij}(R)}{1-S_{ij}^{2}(R)}}=k\eta _{ij}(R),i\neq j;(VIII)$
quae autem formula, integralia superpositionis $S_{ij}$ orbitalium $i$ et $j$, quorum centra $R$ distant, introducens, apte efficit, ut $A_{ij}^{\prime }$ possit aestimari approximate cum $R$ mutatur. Quibus rebus positis, matrix $A^{\prime }$ dimeris (vii) huiusmodi erit:	introducing in which formula the overlap integrals $S_{ij}$ of the orbitals $i$ and $j$ whose centers are separated by $R$ fairly suffice that $A_{ij}^{\prime }$ would be possibly estimated when $R$ varies. Under these assumptions the matrix $A^{\prime }$ of the dimer (vii) will be:
$A^{\prime }=\left|{\begin{array}{cccc}0&\eta (a)&\eta (a+b)&\eta (2a+b)\\\eta (a)&0&\eta (b)&\eta (a+b)\\\eta (a+b)&\eta (b)&0&\eta (a)\\\eta (2a+b)&\eta (a+b)&\eta (a)&0\end{array}}\right|(IX)$
valoribus $\eta (R)$ in tabula I datis [in praesenti instantia, $i$ et $j$ semper 1s orbitalia hydrogeni sunt, unde tantum $\eta (R)$ pro $\eta _{ij}(R)$ scribere licet].	with the values $\eta (R)$ given in Table I [at the present moment $i$ and $j$ are always 1s orbitals of hydrogen so that it is possible to write $\eta (R)$ for $\eta _{ij}(R)$].
Tabula I. Valores elementorum extra diagonalium $\eta$ duorum 1s hydrogeni orbitalium, quorum centra $R$ distant. Unitas energiarum ~ 2.24 eV est [2]; $a$ = 0.74 Å
$R$ $a$ $2a$ $3a$ $4a$ $5a$ $S$ 0.753 0.390 0.166 0.063 0.025 $\eta$ -1.738 -0.460 -0.171 -0.063 -0.025
Sit $b=a$; transformatione $V$ adhibita, matrix $A^{\prime }$ hoc modo mutatur:	Setting $b=a$ and applying transformation $V$, matrix $A^{\prime }$ so changes:
$A^{\prime }=\left|{\begin{array}{cccc}0&-1.7380&-0.460&-0.171\\&0&-1.7380&-0.460\\&&0&-1.7380\\&&&0\end{array}}\right|\rightarrow V^{\dagger }A^{\prime }V=\left|{\begin{array}{cccc}0.879&-1.499&0&-0.045\\&-0.879&-1.863&0\\&&-0.879&-1.499\\&&&0.879\end{array}}\right|(x)$
ubi illa elementa extradiagonalia, quae nulla sunt considerata, minora quam $10^{-7}$ sunt. Si autem $b=2a$, in quadrato extradiagonali matricis transformatae tantum elementum (2-3), quod valet -0.523, a nihilo differt; elementum enim (1-4) circa $10^{-4}$ est. Ut melius effectus transformationis $V$ eiusque significatio pateat, duos calculos comparemus, calculum I ad valores proprios matricis $A^{\prime }$, calculum II ad valores proprios matricis $V^{\dagger }A^{\prime }V$, elemento (1-4) neglecto, perducentem. Serius apparebit (in columnis 2 et 3 Tab. II) maximum errorem tantum 3% esse, et hoc quidem fieri si $b$ et $a$ equales sint.	where those off-diagonal elements which are considered to be zero are smaller than $10^{-7}$. If meanwhile $b=2a$ in the square of the transformed matrix the element (2-3) whose value is -0.523 differs from zero; the element (1-4) is ca. $10^{-4}$. For the effect of the transformation $V$ would be better attainable let us compare two calculations; calculation I of the eigenvalues of the matrix $A^{\prime }$, calculation II of the eigenvalues of the matrix $V^{\dagger }A^{\prime }V$, performed neglecting the element (1-4). Later the maximal error will be seen (in colums 2 and 3 of Table II) to be only 3% and this only to happen if $b$ and $a$ would become equal.
3 Casus Generalis Polymeris Cuiusvis Generaliter polymeres constructum e pluribus monomeribus, e.g. moleculis hydrogeni in recta jacentibus, eodem modo ac (VII) ostendit, nunc consideremus. Transformationes $V_{i}$, si matrices $B_{ij}$ non variantur, eaedem sunt ac illae, quibus in paragrapho precedente usi sumus; attamen impossibile erit, omnes matrices $B_{ij}$ ad formam, quam in (X) habent, perducere. Cuius rei causam in hoc agnoscere licet, quod matrices $V_{1},V_{2},......$ e quibus $V$ consistit, alternate cum duobus matricibus $W_{1}$ et $W_{2}$ coincidunt; unde, ex matricibus $B_{ij}$, $V$ adhibita, $W_{1}^{\dagger }B_{ij}W_{2}$ et $W_{2}^{\dagger }B_{ij}W_{1}$ alternate fiunt, quarum, quamvis $B_{ij}$ semper eadem matrix $B_{12}$ sit, altera tantum formam diagonalem $b_{12}$ habet; quod ex formula sequenti clare apparet :	General Case of Arbitrary Polymer Let us consider polymers generally constructed of numerous monomers e.g. molecules of hydrogen arranged linearly the same way as (VII) shows. Transformations $V_{i}$, provided matrices $B_{ij}$ do not vary. are the same as those we used in the preceding paragraph; otherwise, it will be impossible to bring all matrices $B_{ij}$ to the form they have in (X). For this reason it suits to recognise that matrices $V_{1},V_{2},......$ of which $V$ consists, coincide with two matrices $W_{1}$ and $W_{2}$ in an alternate manner; where from upon appliyng matrix $V$ from matrices $B_{ij}$ <matrices> $W_{1}^{\dagger }B_{ij}W_{2}$ and $W_{2}^{\dagger }B_{ij}W_{1}$ arise since whatever $B_{ij}$ in all cases would be the same matrix $B_{12}$ which <upon transformation> has the diagonal form $b_{12}$ which is clearly seen from the following formula:
$V^{\dagger }A^{\prime }V=\left|{\begin{array}{ccccccc}W_{1}^{\dagger }AW_{1}&W_{1}^{\dagger }BW_{2}\\W_{2}^{\dagger }B^{\dagger }W_{1}&W_{2}^{\dagger }AW_{2}&&W_{2}^{\dagger }BW_{1}\\&W_{1}^{\dagger }B^{\dagger }W_{2}&&W_{1}^{\dagger }AW_{1}&&W_{1}^{\dagger }BW_{2}&\ldots \\&&\vdots &&\ddots \\&&\vdots &&&\ddots \end{array}}\right|$
Duplex igitur monomeres considerandum esse oportet ut concedamus; qua re autem concessa nullo mode casus plurium monomerium a casu duorum differt. Tunc enim per transformationem $V$ matrix $A^{\prime }$ ad formam	Thus a duplicated monomer is to be considered and let us leave <it>; the left <thing> <we see> the case of numerus monomers from the case of two mobomers in no way is different. Thus, by the transformation $V$ the matrix $A^{\prime }$
$A^{\prime \prime }={\begin{array}{cccccccccccc}\\&A_{0}\\&&&r\\&&r\\&&&&A_{0}\\&&&&&&r\\&&&&&r\\&&&&&&&A_{0}\\&&&&&&&&&r\\&&&&&&&&r\\&&&&&&&&&&A_{0}\\\\\end{array}}(XI)$
reducetur, quibusdam parvis elementis ut antea neglectis. Reductioni ad diagonalem matricis $A^{\prime \prime }$ methodus Cl. Vir. Coulson et Rushbroock [1] adhiberi potest; sicut autem quilibet numerus monomerium semper cum eadem forma structuraque matricis $A^{\prime \prime }$ correspondere debet, licet extrapolationem iuxta modum adhibere, ad longioris polymeris energias (id est valores proprios maioris $A^{\prime \prime }$) aestimandas (vide col. III Tab. II). Extrapolationem demum ad graves errores non ducere probari potest infinitum polymeres iuxta methodum Egr. Vir. KOUTECKY [7] tractando, quod etiam ostendet, num revera liceat aliquot elementa matricis $V^{\dagger }A^{\prime }V$ negligere, id est $A^{\prime \prime }$ pro $V^{\dagger }A^{\prime }V$ considerare (vide col. IV Tab. II).	reduces to the <above> form, where the small elements are as previously neglected. For a reduction of the matrix $A^{\prime \prime }$ to the diagonal <form> the method of Coulson et Rushbroock [1] can be applied. Similarly, as an arbitrary number of monomers must always correspond with the same form and structure of the matrix $A^{\prime \prime }$, it is allowed to apply the same way an etrapolation to/for the energies of longer polymers (that is <higher?> eigenvalues of $A^{\prime \prime }$) to be estimated (see col. III Tab. II). This extrapolation can be proven not to lead to severe errors <in the case of> infinite polymers <provoded> the method of KOUTECKY [7] <is applied>.
Tabula II. Energiae orbitales polymeris linearis molecularum hydrogeni secundum varias methodos tractati	Table II. The energies of the linear polymer of hydrogen molecules treated according to different methods.
(1) (2) (2bis) (3) (4) $b=a$ 1.738 2.454 1.379 2.432 1.420 2.694 1.079 2.278 1.091 -1.738 -0.545 -3.289 -0.569 -3.283 0.784 -4.556 0.749 -4.568 $b=2a$ 1.738 1.842 1.658 1.842 1.658 1.920 1.557 1.920 1.557 -1.738 -1.318 -2.181 -1.319 -2.181 -0.874 -2.603 -0.874 -2.604
(1) Energiae orbitales monomeris. (2) Energiae orbitales dimeris exacte tractati (calculus I huius dissertationis). (2bis) Idem (calculus II huius dissertationis). (3) Limina fasciarum energiae infiniti polymeris per extrapolationem tractati (vide aequationem XII). (4) Idem, sec. Adn. [7].	(1) The energies of orbitals of a monomer. (2) The energies of orbitals of a dimer treated exactly (calculation I of the present paper). (2bis) The same (calculation II of the present paper). (3) Limits of the energy bands of the infinite polymer treated by extrapolation (see equation XII). (4) The same, according Ref. [7].
Quas res Tab. II illustrat in simplicissimo casu plurium hydrogeni molecularum. In Tab. II integralia superpositionis neglecta sunt; quae si introduxissemus, fasciae superiores latiores, non angustiores quam fasciae inferiores essent [11]. Ibi specialiter est notandum, quod errores ab absentia parvorum elementorum et ab extrapolatione inducti a numero monomerium pendere non videntur. Infra ostendemus methodum in Adn. [1] propositam ad alia non simplicia polymera posse adhiberi, eodem modo ac ad polymeres molecularum hydrogeni: id est, matrices $B_{ij}$ per transformationem $V$ ad diagonales perductas saepe unum elementum $r=b_{ij}^{\mathrm {max} }$ habere quod multo maius ceteris sit, immo adeo excellat, ut illud tamquam mensuram interactionis monomeris cum monomere possimus interpretari. In polymere generali, sicut diximus, dimeres est considerandum, quocum e matrice $A^{\prime \prime }$ submatrix $A_{d}^{\prime \prime }$ correspondet:	Table II illustrates that all in the simplest case of multiple hydrogen molecules. In Table II the overlap integrals are neglected, which if we them introduce <make> upper <energy> bands wider, <and> not narrower then lower <ones> [11]. Here, it needs to be specifically noted that the errors induced by the absence of minor elements and by extrapolation do not seem to depend on the number of monomers. Below we shall show that the method proposed in Ref. [1] can be applied to other non <so> simple polymers by the same way as to polymers of hydrogen molecules: that is, matrices $B_{ij}$ reduced to the diagonal <form> by the transformation $V$ freaquently have <only> one element $r=b_{ij}^{\mathrm {max} }$ which would be much larger than others, so that it comes to <the extent> that we can interpret them as a measure of interaction of a monomer with a monomer. in a general polymer, as we said, the dimers are to be considered, for which a submatrix $A_{d}^{\prime \prime }$ from the matrix $A^{\prime \prime }$ <does> correspond:
$A_{d}^{\prime \prime }=\left|{\begin{array}{cc}W_{1}^{\dagger }\left(A_{1}+\Delta A_{1}\right)W_{1}&W_{1}^{\dagger }B_{12}W_{2}\\W_{2}^{\dagger }B_{21}W_{1}&W_{2}^{\dagger }\left(A_{2}+\Delta A_{2}\right)W_{2}\end{array}}\right|$
(Hic, sicut diximus, $W_{1}$ et $W_{2}$ tales transformationes sunt ut $W_{2}^{\dagger }B_{12}W_{1}$, certa quadam permutatione adhibita, diagonalis matrix $b_{12}$ sit.) $A_{d}^{\prime \prime }$ nova est descriptio monomeris duplicis in polymere, cui hoc est peculiare, quod eius interactiones cum vicinis monomeribus matrix $b_{12}$ repraesentat, quae ex minimo numero elementorum non nullorum consistit, itaque unice est definita. Physice igitur novam basim, quae ex priore per transformationem $V$ exsistit, licet interpretari tamquam commixtionem orbitalium atomicorum perducentem ad electrice polaria orbitalia unum monomeres simplex tantum involventia, quorum interactiones cum orbitalibus eius monomeris, quod est altera pars monomeris duplicis, sunt complicatae, interactiones vero cum orbitalibus ceterorum monomerium simplices. Ex autem orbitalibus novae basis, unum pro dimere ad interactionem maximam inter dimera ducet; adeoque, si matricis $b_{12}$ unum tantum elementum multum a nihilo differt, interactionem duorum monomerium duobus orbitalibus tantum adscribere licet.	(Here, as we said, $W_{1}$ et $W_{2}$ are such transformations that $W_{2}^{\dagger }B_{12}W_{1}$, with the permutations applied required <to make> the matrix $b_{12}$ diagonal). <The matrix> $A_{d}^{\prime \prime }$ is a new description of duplicated monomers in the polymer, of that specific that their interactions with neigbor monomers would be represented by the matrix $b_{12}$ which consists of <the/a> minimal number of nonvanishing elements, by this uniquely defined. Physically, the new basis, which appears from the previous by the transformation $V$, may be so interpreted that the mixture of the atomic orbitals reduced to the electrically polarized orbitals of one simple/single monomer so <re>combined that their interactions with orbitals of that monomer which is another part of the doubled monomer are complex whereas interactions with orbitals of other monomers are truely simple. Meanwhile, from the orbitals of the new basis one per dimer leads to the maximal interaction between dimers, if only one element of the matrix $b_{12}$ significantly differs from zero. <?>
4 Exemplum Polypeptidum Describamus nunc calculos quos fecimus ad fascias energiae peptidum (v. fig.) determinandas. In his calculis, basim orbitalium π Slaterii aliorum erga alia symmetrice orthogonalium factorum adhibuimus; cum autem coniugatio inter	Polypeptide example Let us now describe calculations which we did for determining the energy bands of peptides (see figure). In these calculations we employed a basis of Slater π orbitals with regard to (?) other symmetrically orthogonal factors. As, meanwhile, the conjugation between the NCO units
figura
unitates NCO, quam Egr. Vir. EVANS et GERGELY consideraverant [5], minime contribuisset energiae catenae, quid maior coniugatio efficeret investigavimus, effectum hydrogenici pontis tractando tamquam si exsisteret ex coniugatione orbitalium 2p hydrogeni [11].	which EVANS et GERGELY had considered [5], had minimally countributed to the energy of the chain, we investigated what the major conjugation produced, the effect of hydrogen bridges treating <them> so as they would arise from conjugation of the 2p orbitals of hydrogen [11].
Tabula III. Valores proprii matricum $B_{12}^{\dagger }~B_{12}$ cum diversis unitatibus correspondentium	Table III. Eigenvalues of the matrix $B_{12}^{\dagger }~B_{12}$ corresponding to different units
NCOH HNCO COHN OHNC 27.99 19.03 20.00 44.78 2·10-2 2·10-2 10.75 11.19 2·10-3 2·10-3 8·10-4 2·10-4 4·10-8 1·10-8 2·10-5 3·10-7
Si elementa matricis Hamiltonianae ad basim supra definitam relatae methodo MO-SCF calculamus, catenam quattuor monomerium considerantes, matrices centrales tanquam matrices monomerium "in situ" eorumque interactionis in longissima catena sumi possunt. Nova tamen quaestio hic apparet, quae inter unitates NCOH, NHCO, COHN, OHNC tamquam monomeres sit consideranda. Ut eam solvamus, matrices $B_{12}^{\dagger }~B_{12}$ cum quattuor casibus correspondentes ad diagonales secundum aequationem (II, bis) reducamus: elementa matricum $(b_{12})^{2}$ ita obtentarum Tab. III indicat; e qua apparet unitatem NCOH seligendam esse, si unum tantum elementum sit in $b_{12}$ servandum. Tunc enim id elementum matricis $b_{12}$, quod maximum valorem absolutum habet, $b_{12}^{\mathrm {max} }=-5.291$ eV est; cetera autem elementa eiusdem matricis non maiora sunt absolute quam illae inter non vicinas atomos interactiones, quas omnes solent negligere. Tab. IV, tandem, quod ex calculo efficitur monstrat. Calculus autem ita perductus est. Methodus citata [1], ad matricem (XI) adhibita, dat inaequationem:	If we calculate the elements of the Hamiltonian matrix in the above defined basis by the MO-SCF method, considering a chain of four monomers, the central <block matrices> of such matrix can be accepted (?) as matrices of monomers "in situ" as well as of the interactions thereof in the longest chain. Here meanwhile a new question appears, which among the units NCOH, NHCO, COHN, OHNC is to be considered as monomer. In order to solve it let us reduce the matrices $B_{12}^{\dagger }~B_{12}$ corresponding to the <above> four cases to the diagonal <form> according to eq. (II, bis). Tab. III shows the elements of the matries $(b_{12})^{2}$ so obtained, from which it is evident that the unit NCOH is to be selected, if <only> one element would be conserved in $b_{12}$. Thus, that element of the matrix $b_{12}$, which has the maximal absolute value is $b_{12}^{\mathrm {max} }=-5.291$ eV; other elements of the same matrix which are not larger by absolute vlaue than that <describe> interactions between nonneighbor atoms which all are customarily neglected. Table IV meanwhile shows that which comes for the calculation. The calculation meanwhile is so performed. The cited method [1] as applied to matrix (XI) gives an inequality:
$(a-r^{2}d^{2})^{2}-4r^{2}(bc-ad)<0$, (XII)
ubi $a$ est determinans matricis $A_{0}-eI$, valoribus propriis e vocatis;	where $a$ is the determinant of the matrix $A_{0}-eI$ with $e$ being the eigenvalues.
Tabula IV. Energiae orbitales polypeptidum. Vide etiam Adn. [11] et Tab. II	Table IV. The energies of the polypeptide orbitals. See also Ref. [11] and Tab. II
1 2 3 4 +6.087 +14.110 +2.477 +14.117 +2.441 1.244 +1.595 +0.405 +.1258 -0.343 +1.294 -0.345 -12.681 -12.227 -13.214 -12.495 -12.719 -12.456 -12.786 -15.052 -14.815 -15.771 -15.081 -15.989 -15.036 -16.014
(1) Energiae orbitales monomeris. (2) Energiae orbitales dimeris exacte tractati (calculus I huius dissertationis). (3) Limina fasciarum energiae infiniti polymeris per extrapolationem tractati (vide aequationem XII). (4) Idem, sec. Adn. [7].	(1) The energies of orbitals of a monomer. (2) The energies of orbitals of a dimer treated exactly (calculation I of the present paper). (3) Limits of the energy bands of the infinite polymer treated by extrapolation (see equation XII). (4) The same, according Ref. [7].
$b,c,d$, determinantia sunt earum matricum, quae ex $A_{0}$ exsistunt tum, cum vel prima linea primaque columna, vel ultima linea ultimaque columna, vel utraeque lineae columnaeque supprimuntur. Ex inaequatione (XII) valores proprii $e$, qui limina sunt fasciarum energiae, inveniri possunt, aequatione gradus $4n$ resoluta (quod minimis erroribus fieri potest, postquam $A_{0}$ ad diagonalem reducta est). Possumus nunc quaestionem solvere, quomodo ex dipeptide ad polymeres infinitum energiae orbitales varientur, et quid absentia parvorum elementorum matricis $b_{12}$ efficiat, energias per nostram methodum obtentas cum energiis obtentis per methodum in Adn. [7] propositam comparando, eodem modo ac in Tab. II pro hydrogenicis catenis fecimus. In Tab. IV plures casus comparantur: calculi ad pauca polymera spectantes, vel directe vel secundum formulam (XII) perducti; calculi ad infinita polymera spectantes, vel secundum (XII) vel secundum Adn. [7] perducti. Comparatio ostendit neque absentiam parvorum elementorum quae neglecta sunt, neque extrapolationem ad magnos errores ducere. Exemplum peptidicae catenae proprietates utilitatemque methodi hic propositae bene illustrat; quae praecipue in hoc consistunt, quod, illa methodo, perturbatio valorum energiae monomeris a proximitate aliorum monomerium inducta velut functo unius quantitatis $b_{12}^{\mathrm {max} }$ tractari potest, quamvis non sine approximatione, at quod generalius simplicissima forma matricis interactionis monomerium cum monomeribus praeberi potest. Quid autem physice status monomeris in situ, transformatione $V$ adhibita, significet, et quid investigatio mutationis energiarum, interactione mutante, possit docere, alio loco, aliis certis exemplis dicemus, non desperantes fore ut lectores ipsos huius dissertatiunculae propositiones ex exempla, quamvis minima, ad novas inventiones stimulent.	$b,c,d$, are determinants of those matrices which arise from $A_{0}$ provided the first row and the first column, or the last row and the last column or whatever rows and columns are erased. From eq. (XII) the eigenvalues $e$, which are limits of the energy bands, can be derived, with a resolved equation of the power $4n$ (which can be done with a minimal error after $A_{0}$ being reduced to the diagonal <form>). Now we can solve the question, how the energies of the orbitals vary <while going> from the dipeptide to the infinite polymer and how the absence of the smaller elements of the matrix $b_{12}$ would effect, by comparing the energies obtained by our method with the energies obtained by the method proposed in Ref. [7] following the same way we did in Table II for the chain of hydrogens. In Tab. IV more cases are compared: those calculations performed for small polymers either directly or according to formula (XII), those performed for infinite polymers either according formula (XII) or according to Ref. [7]. The comparison shows neither the absence of of small elements which are neglected nor the extrapolation to lead to big errors. The example of the peptide chain nicely illustrates/highlights properties and utility of the method proposed here, which mainly consist in that that by this method the perturbation of the energy values of the monomers induced through the presence of other monomers can be treated just as a function on a single quantity $b_{12}^{\mathrm {max} }$, although not without an approximation, and that in general a very simple form of the matrix <describing> interaction of a monomer with <another> monomer can be derived. What the status of a monomer in situ under the applied transformation $V$ physically signifies and what the study of the energy variations with the variation of interaction could teach, we shall say in other place with other clear examples, not being desperate that the conclusions <derived> from examples, although minimal, of the present paper, will stimulate the readers themselves for new discoveries.
Gratias agimus Cl. Vir. RAPHAELI DEL RE, qui nos, quod ad usum linguae latinae attinet, consilio benigne adiuvit.	We thank RAPHAEL DEL RE who to us, whom he brought to usage of Latin, kindly helped with advises.
Adnotationes	References
[1] COULSON,C. A., et R. S. RUSHBROOKE: Proc. Cambridge philos. Soc. 44, 272 (1948). [2] DAUDEL, R., R. LEFEBVRE, et C. MOSER: Quantum Chemistry. New York: Interscience Publishers 1959. [3] DEL RE, G.: Theoret. chim. Acta (Berl.) 1, 188 (1963). [4] DEL RE, G.: Electronic Aspects of Biochemistry, p. 227. New York: Academic Press 1964. [5] EVANS, M.G., et J. GERGELY: Biochim. biophys. Acta 8, i88 (1949). [6] FORCELLINI, A.: Monomeres et polymeres: Totius Latinitatis Lexicon, p. 168, Prati 1868. [7] KOVTECKY J., et R. ZAHRADNIK: Coll. Czechoslov. chem. Commun. 25, 811 (1960). [8] LADIK, J.: Acta physica Acad. Sci. hung. 15, 287 (1963). [9] MULLIKEN R. S.: J. Chim. physique 46, 675 (t949). [10] SUARD, M., G. BERTHIER, et B. PULLMAN: Biochim. biophys. Acta 52, 254 (1961). [11] SUARD,M.: J. Chim. physique 62, 89 (1965).
Prof. Dr. G. DEL RE Gruppo Chimica Teoriea del CNR Via Cornelio Celso 7 Roma, Italia