object, Part 2: Molecular derivation and application to

This study, based on mere considerations induced by the Special Theory of Relativity, has previously established the following relationship between the “minimum electronic energy” , and the related “classical vibration frequency” , in regards to electronic states of a given diatomic molecule: . Where  is the reduced mass of the molecule,  the “internuclear distance” associated with , and  a Lorentz invariant dimensionless coefficient, insuring the equality; it depends only on the electronic structure of the molecule; therefore for electronic states configured similarly, we expect the coefficient , to remain practically the same; it takes values, roughly around unity. The framework in question is interesting, given that, for alike electronic states of a given molecule,   versus , should behave linearly. This further, should allow the determination of, for the states in consideration. The expression is anyway valid for any diatomic molecule, along with a given . On the other hand, the “ground states” of bonds delineating chemical similarities, display “alike electronic configurations”. This means that,  for such bonds, should remain practically the same. Thus, regarding the ground states of such molecules,  versus  should further be expected to behave linearly (the quantities of concern, now being exclusively assigned to the ground states of the molecules in question).  We check this prediction successfully for the entire body of diatomic molecules and calculate , for different “chemical families”. The relationship we discover has got as much predictive power as that provided by the classical quantum mechanical tools; it is though incomparably simpler and faster. 
 
   
 
 Key words: Special theory of relativity, quantum mechanics.

The interdependence of electronic energy, period of time, mass and internuclear distance displayed by any object, Part 2: Molecular derivation and application to chemically alike molecules This study, based on mere considerations induced by the Special Theory of Relativity, has previously established the following relationship between the "minimum electronic energy" min E , and the related "classical vibration frequency" ω, in regards to electronic states of a given diatomic molecule:

INTRODUCTION
In this study, it has been shown earlier that the special theory of relativity (STR) imposes the following.Already at rest, the "period of time" 0 T , the "characteristic length" 0 R , the "clock mass" 0 M , to be associated with the "internal dynamics" of an atomistic or molecular quantum mechanical object, and the "electronic energy" 0 E , *Corresponding author.E-mail: mansursah@gmail.com.
It can be checked that the proportionality constant, coming into play, is dimensionless; the somewhat new denominations introduced here, have been defined earlier.
Equation (1) outlines the architecture delineated by any given entity with respect to "period of time" 0 T , "characteristic length" 0 R , "clock mass" 0 M , and "total energy" 0 E , one can associate with the "internal dynamics" of an atomistic or molecular quantum mechanical object.It can be used right away, for a set of entities, for which the dimensionless proportionality constant it involves, can be considered the same.This is the heart of the presen approach.
As stated earlier, we have arrived at Equation (1), on pure relativistic considerations,.Note then that the proportionality constant coming into play is determined to be necessarily a Lorentz invariant constant.In this study, we are going to provide a direct derivation of Equation( 1), for diatomic molecules, essentially on a quantum mechanical basis, and show that the proportionality constant it involves, indeed depends only on the electronic configuration of the bond coming into play.Thus, for a set of molecules embodying electronically alike bonds, we should expect 0 T to vary linearly with respect to ; this behavior yields the proportionality constant of the set of concern.
In this study, we are embarrassed to refer mainly to our previous work.The fact remains that we found practically nothing similar to the line we pursue herein.Subsequently, we will first derive Eq.( 1), on a quantum mechanical basis (Sections 2 and 3), after which we draw a conclusion.

DERIVATION
Let E(R) be the electronic energy of the diatomic molecule of concern, at a given state, with respect to the internuclear distance R (Herzberg, 1964).This is a quantum mechanical output, of course.Thus, E(R) can be as usual defined as the eigenvalue of the Schrödinger equation describing just the electronic configuration of the molecule, with fixed nuclei, within the frame of the Born and Oppenheimer approximation (Born and Oppenheimer, 1927).E(R), following Morse's parabolic approximation (Morse, 1929), can be expressed, in terms of the "force constant" k , associated with the classical vibration frequency ω (Davis, 1965), as: (2) (Morse approximation to the electronic energy, versus Yarman et al. 4479 the internuclear distance) It should be emphasized that this relationship, does not display characteristics such as "anharmonicity" and "dissociation".Nevertheless, at a given ground or excited electronic state, Equation (2), will remain very satisfactorily correct, in regards to the lowest vibrational levels, taking place at this state, given that the Morse's parabolic approximation, is particularly valid, around min E , and that we will specifically focus on the ground vibrational level at the electronic state of concern.
The parabola, approximately describing E(R) (which ought to be anyway a negative quantity, for it represents the total electronic energy of the molecule), intercepts the R axis at 00 R , which we can define with respect to min R , as: (3) [Hypothetic internuclear distance which makes the approximated E(R), vanish] Note that this is just a definition.Note also that, there are two roots.These do not mean anything physically, since the Morse parabola is just an approximation.Yet it is still helpful to proceed with Equation (3).It is just a trick, to transform Equation (2), as we will soon make use of.The parameter p, is unknown, but we do not really have to determine it.In any case, it is perfectly legitimate to write Equation (3).
Equations ( 2) and (3), provide us with the possibility of expressing min E , as: (Expression of the minimum energy in terms of the force constant and the minimum internuclear distance).
Next we define a new quantity k g as: 2 (5) A quantum mechanical elaboration on k g is provided in Yarman (2004c).Anyway, we expect k g to remain practically the same, for electronic states configured similarly.We can thus write Equation (4) as: Thus, using the familiar expression for the classical vibration frequency (Davis, 1965); (Classical vibration frequency) In Equation ( 4), we can finally arrive at the relationship: [Equation (1) obtained, via quantum mechanical considerations regarding electronically excited states of a given molecule] Given that the frequency is the inverse of the period of time, this relatioship is, in fact, identical to Equation (1).
We have arrived to it, on the basis of a given electronic state of the molecule at hand.Thus it may be considered for all electronic states of this molecule.Thus, we can right away conjecture that the plot of , should come out, as a straight line with regards to the electronic states of the given molecule, which would configured similarly.Yet the framework of Equation ( 8) is more general than this, for it is valid for any molecule, and any state of a given molecule.And it is interesting to consider it, when the given states are configured similarly.There are two distinct interesting sets of these.The first one, in effect, consists in the electronic excited states of a given molecule, configured similarly.That is, even if the molecule is at an excited state, the bond configuration, may remain about the same, and the coefficient g k could then be considered as a constant.This is one thing.The second set in question, consists in the ground states of a given type of molecules, whose bond electronic configurations, are alike.This option will be considered particularly in the later part of the study.
The important thing about Equation ( 8) is that, it embodies no universal constant, say the Planck constant, or electric charges.It is a unique relationship, in that sense, made of four fundamental quantities, that is, mass, size, period of time (or the same frequency), and total energy.It universally displays the interrelation in between the quantities in consideration.
Recall that, we arrived at it earlier, based on mere considerations, just within the frame of the STR.In that sense Equation ( 8) is most general, and should hold at any level, for mass, size, period of time, and total energy, associated with that level, and it becomes interesting for any set of entities for which the proportionality coefficient g k may be conjectured to remain the same.Though we restrict ourselves here to the study of Equation ( 8), with regards to diatomic molecules only.

GROUND STATES OF DIATOMIC MOLECULES BELONGING TO A GIVEN CHEMICAL FAMILY
Equation (8) (as shown earlier), is in fact valid for any electronic state of any given molecule.Thus, it is surely valid for the ground state of the molecule of concern.Then it becomes particularly interesting for the ground states of molecules configured similarly.The ground states of diatomic molecules belonging to a given "chemical family", constitute such states.In different terms, a chemical family embodies molecules, whose bonds are configured similarly.Molecules made of alkali atoms, for instance, constitute a chemical family.
The idea remains that, the proportioanlity constant of Equation ( 8) can be considered as a constant, for the set of molecules one tackles with.Let then 0 E be the magnitude of the ground state electronic energy, 0 ϖ the classical vibration frequency at 0 E , and 0 R the ground state internuclear distance of the molecule of concern.With these definitions, Equation (8) becomes: [Equation (1) obtained, via quantum mechanical considerations regarding the ground state of a given molecule] Thus, we conjecture that the plot of , should come out as a straight line for the slope of which will furnish the coefficient k g .
We can equally write: (Relationship we derived quantum mechanically and we consider for the ground states of molecules belonging to a given chemical family, where the electronic configurations of the bonds can be considered to remain the same) where 0 T is the inverse of 0 ϖ ; this is obviously Equation ( 8), where though the proportionality constant is specifically displayed.This equation suggests that, for chemically alike molecules, the period of vibration 0 T , should behave as proportionally to . The reason we originally considered Equation (10), instead of Equation ( 9), is simply that, it expresses the elementary quantity of "period of time", in terms of the elementary quantities "space" (size), and "mass" (reduced mass), next to "energy".As explained through the Introduction earlier, the frame displayed by Equation ( 10) is Lorentz invariant.
The fact that the mentioned quantities are structured in the given way, insures the end results of the STR, were the molecule brought to a uniform translational motion (Herzberg, 1964;Born and Oppenheimer, 1927), or the end results of the General Theory of Relativity, were the molecule embedded in a gravitational field, in fact any field the object may interact with, and this is how originally, we have arrived straight at Equation (10).
We have calculated the magnitude of the electronic energy, at the ground state of the diatomic molecule AB (made of the atoms "A" and "B"), in consideration, from the relationship: display our prediction for all of the diatomic molecules, reassembled on the basis of their chemical similarities (Davis, 1965;Spirko et al., 1993;Diemer et al., 1984).Data used to draw these figures are reassembled in Tables 1 to 11 (Herzberg, 1991).For each chemical family of concern, k g is calculated and presented in Table 12.

CONCLUSION
In the earlier part of this work, we had derived Equation (1) within a most general framework, through simple considerations, based on just the STR.Here, to have a cross check, we have derived this relationship, for a diatomic molecule, based on simple quantum mechanics, and landed at Equation ( 10).The advantage of it is that, it does not embody any universal constants (such as the Planck Constant, for instance); in other terms, it is free of any such constants.It is based on four fundamental           the same for a given chemical family, and varies between, approximately, 0.3 and 1, with regards to the entire body of the chemical families we have considered.

Molecules
One can show that g k happens to be roughly, inversely proportional to the dissociation energy of the molecule (Yarman, 2004c).It is indeed easy to see that, the greater p (Equation 3), the smaller is the dissociation energy.Thus, as one can observe from Table 12; g k decreases as the bond becomes stronger.This is why we would like to call it bond looseness factor (Herzberg, 1964) Briefly speaking, the higher, the number of the covalent bonds, making the overall bond of the diatomic molecule, the smaller will g k be.Or the smaller the number of free electrons an atom possesses, the looser will be the bond it will make, with say, a halogen atom, thus the higher will g k be, etc.
At any rate our goal here, was not really to disclose just a molecular characteristic; the property (Equation 10), we came to unveil was in fact a mean to show that matter, more specifically molecules, ought to be built in a unique manner in order to insure the end results of the STR,  were the object at hand, brought to a uniform translational motion, or those of the General Theory of Relativity, were the object, embedded in a gravitational field, in fact in any field it interacts with.It is henceforth question of a distinct architecture framing how the main quantities, that is, mass (clock mass), space (characteristic length), time (period of time), and energy (base of the clock labor in quesiton, carried by the clock mass), painting the internal dynamics of the object in hand, are structured, in regards to each other.

Average of
Along this line, one can write Equation (1), even without any knowledge of quantum mechanics, at once, just the way we did in the earlier part of this work.This, in effect, represents a Lorentz invariant cast (as we have shown in the earlier part of this work).Thus, it is nothing else, but a matter of intuition, to write it down.
Testing it, is further, a matter of an easy exercise, given that the classical vibrational period of time 0 T , versus the quantity , should behave linearly, were the proportionality constant coming into play, remained the same for a selected set of molecules, which indeed happens to be the case for molecules belonging to a given chemical family.
This is what we have shown in this article, which would have otherwise turned out to be a mathematical impossibility, throughout much too cumbersome quantum mechanical setups; our approach both relativistically and quantum mechanically furthermore, seem to bear, as much predicitive power, as that of the corresponding classical quantum mechanical setups.
Let us mention the following, for those who are not much familiar with the usual spectroscopic practice: In our plots, the dimension of the classical vibrational period T 0 is provided in "cm" unit; in other words, a period that would have normally been expressed in "seconds", is here, multiplied by kc, that is, the "specified number" k × the "speed of light" expressed in cm/second; thus to transform the unit of the given classical vibrational period, back to "second-unit", one has to divide it by kc.
invariant dimensionless coefficient, insuring the equality; it depends only on the electronic structure of the molecule; therefore for electronic states configured similarly, we expect the coefficient k g , to remain practically the same; it takes values, roughly around unity.The framework in question is interesting, given that, for alike electronic states of a given molecule, linearly.This further, should allow the determination of k g , for the states in consideration.The expression is anyway valid for any diatomic molecule, along with a given k g .On the other hand, the "ground states" of bonds delineating chemical similarities, display "alike electronic configurations".This means that, k g for such bonds, should remain practically the same.Thus, regarding the ground states of such molecules,
for members of a given chemical family, indeed holds satisfactorily; this constancy C, turns out to be k g 2π C = ; where Å stands for Angstrom, and c for the speed of light.More specifically, k g , stays indeed fairly

Table 4 .
R Data for Figure4.

Table 12 .
The constant k g for different families.