Dedicated to the Memory of

Professor G. Basov[*]

 

AN ESSENTIAL APPROACH

TO THE ARCHITECTURE OF DIATOMIC MOLECULES ­

 

Tolga Yarman

 

Iºýk University, Maslak, Istanbul, Turkey

 

 

ABSTRACT

 

We consider the quantum mechanical description of a diatomic molecule of “electronic mass” m_0e , “internuclear distance” , and “total electronic energy” E_0e. We apply to it the B&O Approximation, together with the cast ~h² (we established previously), written for the electronic description (with fixed nuclei). Our approach yields an essential relationship for T₀, the fundamental vibration period, at the total electronic energy E_0e , i.e. T_{0 =} , where  is the reduced mass of the nuclei, m_e the mass of the electron, g a dimensionless and relativistically invariant coefficient; roughly around unity; it is a quantity associated with just the electronic structure in consideration, thus remains practically the same for bonds bearing similar electronic configuration; n₁ and n₂, are basically the principal quantum numbers of electrons making up the bond(s) of the diatomic molecule in hand. This relationship holds generally. It essentially yields T₀~, for the fundamental vibrational periods versus the square of the internuclear distance at different electronic states of a given molecule, which happens to be an approximate relationship known since 1925, but not disclosed so far. For electronic states configured similarly, we determine n₁n₂ to be R₀/R₀₀, where R₀ is the internuclear distance at the given electronic state, and R₀₀ the internuclear distance at the ground state. This further allows us to draw a complete systematization of diatomic molecules, given that g (appearing to be purely dependent on just the electronic structure of the molecule), stays constant for chemically alike molecules, and  can be identified to be  for diatomic molecules, whose bonds are electronically configured in the same way,  then being the internuclear distance of the molecule with the fundamental ground state vibrational period, picked up as the reference molecule within the chemical family in consideration.     

 

 

 

 

 

 

 

This work is issued from a totally different perspective than the one considered herein. We are not going to reinforce this substantial perspective through this article. Nevertheless we should state it briefly, since it allowed us, long ago to derive practically everything we present herein. [1]^,[2],[3] Thus it was the author’s original idea that, in order to insure the validity of the theory of relativity, in any entity existing in nature, the “architecture of the internal dynamics” this displays, ought to be made in just a given manner.

 

 

In effect any natural entity has got an internal dynamics. Thus it works as a clock. It bears a clock period T₀; the mechanism in question involves a given mass M₀, which we call the “clock mass”, and is installed in a space of size R₀. The “clock mass”, as we shall see, is not a trivial quantity; nonetheless it is not the “total mass” of the entity in hand. One can define several clocks masses, for the same entity, regarding different internal dynamics this displays.

 

The clock mass of the electronic motion of a diatomic molecule, for instance, is the electronic mass, which can be expressed as (a coefficient) x (the electron mass), or just the electronic mass m_e, were the coefficient of concern accounted for, in a different way. On the other hand, the clock mass of the vibrational motion of a diatomic molecule is , where  is the reduced mass of the molecule.

 

Now, the Lorentz transformations on T₀, M₀ and R₀, were the object brought in a uniform translational motion, or similarly, the transformations that these quantities would undergo, were the object embedded in a gravitational field, impose that there ought to be already an intrinsic relationship between T₀, M₀ and R₀, which turns out to be T₀ ~ M₀.^1,2,3,4 This was our original idea, which we will not stress any further, here.

 

In this article though, we will present a derivation of the relationship we conjectured, between T₀, M₀, and R₀, chiefly for diatomic molecules, through the Born and Oppenheimer (B&O) approximation, and a fundamental cast we have derived previously, which we shall briefly sketch (Sections 1, 2 and 3). Then we will elaborate on the quantum numbers that come into play (Section 4). Our approach yields the disclosure of an empirical relationship established back in 1925 (Section 5). Thus we conclude with a novel systematization of all diatomic molecules (Section 6).

 

1. THE UMA CAST

 

For an atomistic or molecular wave-like object existing in nature, we have shown elsewhere the following assertion, first, on the basis of the Schrödinger Equation, as complex as this may be, then on the basis of the Dirac Equation, whichever may be appropriate, in relation to the object in hand.[4]

 

 

 

 

 

 

 Assertion 1:                                                                                                                In a “real wave-like description” composed of I electrons and J nuclei, if the (same) electron masses m_i0, i = 1,..., I and in general different nuclei masses m_j0, j = 1,…, J, involved by the object, are overall multiplied by the arbitrary number , then concurrently, 1) the total energy E_0k associated with the given clock’s motion of the object is increased as much, and 2) the size  of the object in which the given clock’s motion takes place, contracts as much; in mathematical words this is        

 { [(m_i0, i = 1,..., I)  (m_i0, i = 1,..., I) ],  [ (m_j0, j = 1,…, J)  (m_j0, j = 1,…, J) ] }

       Þ { [],  [] } .                                                                 (1)

By “real” we mean, not “artificially gedanken”; for atomistic and molecular wave-like objects, “real object” means, an object embodying a potential energy made of just Coulombian potentials.

 

If the object is, say an atom, then  is the radius of it; if the object is a diatomic molecule,  is the internuclear distance, etc.

 

The occurrence stated by Eq.(1) further yields an invariance, interestingly strapped to the square of the Planck Constant, h².

 

This is the content of our Assertion 2, restated right here.

                                                  

Assertion 2: The quantities,  (k=1,..., K) (associated with the k^th internal motion of the wave-like object in hand), are invariant in regards to a “mass change”, and are all strapped to h².

 

Thus the grand total energy E_{0(GrandTotal)} becomes

 

                                                                      (2)    

 

We call this occurrence, the UMA (Universal Matter Architecture) Cast.

 

 

 

 

 

 

 

 

Note that primarily, what we do here is in not a “dimension analysis”. Anyhow the occurrence we disclose, would not work (i.e. , for the given clock’s motion, would not be invariant in regards to a mass change), if the wave-like object in hand is not “real”, though of course, there still would be no problem, dimension-wise.

 

Soon we shall figure out that the proportionality constant embodied by Eq.(2), besides a usual geometry factor and quantum numbers, fortunately, is made of a “transferable constant”; indeed this constant seems to depend on just the electronic configuration of the molecule. Therefore:

 

i)          It remains the same regarding the electronic states of a given molecule, provided that these states are electronically configured similarly.

 

ii)        Furthermore, it stays the same, regarding the ground electronic states of molecules belonging to a given chemical family, bearing similar electronic configurations.

 

Below we are going to provide a direct derivation of Eq.(2), mainly for the electronic motion of a diatomic molecule, based on the Schrodinger description of it.

 

2. THE B&O APPROXIMATION

 

The quantum mechanical description of a diatomic molecule can be achieved via the usual Schrödinger Equation, involving the “two nuclei” and the surrounding “electrons”. This equation, through B&O approximation, is reduced into the separate descriptions of the “nuclear” and “electronic” motions. We thus come to solve separately the two following Schrödinger Equations, written with the usual notation[5]:

 

       ,                          (3)

 

       .       (4)

 

Here “A” and “B” designate the nuclei, and “e” designates the electrons. We have then the following familiar notation.

 

 

 

 

 

 

 

 

 

mA   mB ZA ZB    me    e

: mass of A : mass of B : atomic number of A : atomic number of B : electron's mass : electron's charge

rAi rBi rii’ rAB

: ith electron's distance to A : ith electron's distance to B : distance between the ith and the i’th electron : distance in between the nuclei : eigenfunction associated with the molecule : eigenvalue associated with the molecule

 

Eq.(3) describes the nuclei vibrational motion, about the internuclear distance  to be input to this equation (for a given electronic state of the molecule), whereas Eq.(4) describes the electronic motion around the two “fixed” nuclei.  is the eigenvalue of the system vibrating around , which as well may rotate; E_e is the electronic eigenvalue, which is in fact the electronic energy of the system whose nuclei are at a fixed distance  from each other. Thus, as usual, one solves Eq.(4), for a given electronic state, in order to determine how the electronic energy E_e varies with respect to , and find the internuclear distance , which makes minimum the eigenvalue E_e, more precisely E_e(r_AB); we call r_ABmin  and E_emin, respectively, the internuclear distance and the eigenvalue in question (for the given electronic state); this is then r_ABmin as , to be input to Eq.(3). Normally E_emin   is negative; yet below, by E_emin we shall mean |E_emin|.

 

The constant  to be input to Eq.(3) is given by

 

                                        =   .                                               (5)

 

                                        [to be determined out of Eq.(4)]

 

Knowing  and  related to the ground electronic state of the diatomic molecule in hand, one can subsequently construct Eq.(3), and solve it as usual, for the vibrational, also rotational eigenvalues E_A,B, associated with the electronic state of the molecule of concern.

 

E_A,B becomes,  

 

                           E_A,B =                 (6)

 

 

I_AB is the “moment of inertia” of the nuclei:

 

                                          I_AB = M_AB  r,                                                                 (7)

   

 

 

 

where M_AB is the nuclei reduced mass.

 

 is the classical vibrational frequency of the molecule, the inverse of which,   T_A,B , is the classical vibrational period of the molecule:

 

                                           T_A,B = 2.                                                              (8)

 

[written on the basis of Eq.(3),

                       were  determined on the basis of Eq.(4)]

 

Thus, along this definition, E_A,B [as expressed by Eq.(6), above], is the solution of Eq.(3), for the nuclear motion of the molecule.

 

3. THE “VIBRATION PERIOD”, VERSUS THE “DIATOMIC MOLECULE CLOCK MASS” AND THE “INTERNUCLEAR DISTANCE”

 

The B&O approach, together with the UMA Cast, stated above, i.e. Eq.(2), allows us to draw an elegant relationship for the vibrational motion of a diatomic molecule, in terms of different masses taking part in the internal motion of the molecule, and the   “internuclear distance” coming into play.

 

Thus Eq.(2), i.e. ~ h², must hold on the basis of Eq.(4); this equation indeed embodies a potential energy term strictly made of Coulombian potential energies. The eigenvalue E_e [more precisely E_e(r_AB)], assumes the value E_emin  when r_AB takes the value of r_ABmin. Furthermore, the only mass that comes into play in Eq.(4), is the electron mass, m_e; in other words, the “clock mass” in question to be associated with the electronic motion of the molecule (with fixed nuclei), is made of only the electron masses coming into play, obviously all bearing the mass m_e.

 

Thence

                    E_emin m_e~  h²    [Eq.(2), written within the frame of Eq.(4)] .    (9-1)

 

The proportionality constant is made of i) a geometry factor, ii) appropriate quantum numbers to be associated with h², and finally iii) a dimensionless, and relativistically invariant quantity that will insure the equality; we will call this quantity , thus associated with the invariance (energy)(mass)(size)², underlined by Assertion 2.

 

 

 

 

 

The quantity  is a  characteristic of the electronic structure;[†] at this step, we rewrite Eq.(9-1), as   

E_emin m_e~  h².                                                                    (9-2)

 

The validity of Eq.(9-2) is checked elsewhere.[6] Nevertheless the check of our end results derived via Eq.(8), should already constitute a “ proof” of it.  

 

E_e() can be as usual expressed fairly in terms of the force constant , defined by Eq.(5), as

                            E_e() =  E_emin +  (– r_ABmin)² .                                      (10)

 

It is true that this relationship does not display characteristics such as “anharmonicity” and “dissociation”; but throughout this work we are going to deal only with the ground vibrational level of the states of concern. Thus, even when we deal with an excited electronic state, Eq.(10) turns out to be quite valid for the ground vibrational level of it.

 

E_e() vanishes at the abscissa , which we can define with respect to r_ABmin, i.e.

 

 = p r_ABmin  [value which makes E_e(), vanish] ;                       (11)

 

p is an unknown parameter at this stage, though it appears to be roughly 2.

 

Eqs.(10) and (11), provides us with the possibility of expressing E_emin, as  

 

                     E_emin = (p - 1)²  .                                                   (12)

 

We plug the RHS of this equation in Eq.(9-2); next we use Eq.(8) to eliminate the force constant ; thus we arrive at the simple expression for , i.e.

 

          ,                                                   (13)

 

where g_k replaces (p-l)²/2.[‡]

Below for simplicity, we call , ; ,;M_AB, M₀, and , .

 

The quantity

                                 M₀ = =                                                          (14)

 

formulated on the basis of the electron mass, has the dimension of a mass. We call it the “vibrational clock mass” (to be associated with the vibrational motion of the diatomic molecule in hand).

 

The proportionality constant drawn by Eq.(13) shall then embody a geometry factor, and quantum numbers. A geometry factor of 2originates from the use of Eq.(9) [where h² may be read as h²/4, and accordingly, 2is left after the square rooting, on the way to Eq.(13)]; an other 2factor originates from the use of Eq.(8); thus altogether, a geometry factor of  4should multiply Eq.(13).

 

The quantum numbers to be introduced in Eq.(13) appear to be more peculiar, and we elaborated on it, as summarized below. Nonetheless, one can sense that [h²] in Eq.(9), should be in fact read as usual, as [n² h²], more precisely as [n₁n₂ h²], n₁ and n₂ being principal quantum numbers of electrons making up the bond(s) of the diatomic molecule in hand³.

 

Recall that because of quantum defects, n₁ and n₂ are not integer numbers.

 

Eq.(13), thus becomes

 

       ,                                                       (15)

 

where now g overall dimensionless, and relativistically invariant quantity, replaces .[§]

 

 

Note here that, the quantum numbers n₁ and n₂ are not necessarily associated with excited states of a given ground state for a given molecule in consideration; we also propose to associate them, with respectively, the ground states of members of a given chemical family, in reference to a given member of this family, more precisely the one possessing the lowest vibrational period; we shall soon work them out.

 

Eq.(15), though g is not known beforehand, is somewhat rigorous. In other terms, despite the B&O approximation we adopted, also the approximate Morse potential we introduced at the level of Eq.(10), the use of g (to be determined), ultimately insures the equality of Eq.(15).

 

It becomes apparent that, g is necessarily related to the electronic structure of the molecule’s bond; thus, for alike bonds, in a given chemical family, we come to expect g to be virtually the same; we call g the “molecular bond looseness factor”, for the inverse of it somewhat characterizes the strength of the bond of concern [cf. the footnotes elaborating on Eqs. (9-2) and (15)].

 

Our approach allows us to draw a whole new systematization of diatomic molecules, and more, such as and the elucidation of an empirical relationship known since long ago, as well as H₂ irregular spectroscopic data. [7]^{, [8]}

 

The introduction of the product of quantum numbers, n₁n₂ requires a demonstration, and that is what we undertake briefly right below, primarily on the basis of the H₂ molecule spectroscopic data.

 

Yet Eq.(15) is worth to be analyzed, even before the elaboration of quantum numbers. Indeed already the plots of  versus , for members of a given chemical family, exhibit nicely increasing, almost faultless, smooth curves; we present eight examples in Figures 1 -  7.

 

4. ELABORATION ON THE QUANTUM NUMBERS

 

The presence of quantum numbers in Eq.(15), is right away induced by the identification of the RHS of Eq.(2) as . This equation is further transformed into Eq.(9-2), written for the mere electronic description of the molecule [cf. Eq.(4)].

 

The excited electronic eigenstates of the molecule should anyway involve quantum numbers.[**] Note further that seemingly not much is known about the quantum numbers, a complex system will assume.

 

 

 

 

The simplicity of Eq.(2) or Eq.(9),  clearly leaves no other room to quantum numbers that shall come into play in these equations, other than that, right next to .

Thus a composite quantum number N (i.e. the product of the two principal quantum numbers to be associated with the bond electrons, in the case of a diatomic molecule),  should come to multiply h², in Eq.(2) or Eq.(9), regarding an excited eigenstate, in just the same way the square of an integer quantum number related to an excited state of the simplest wave-like objects (for example, the hydrogen atom), comes in a similar relationship, to multiply h².

 

This piece of information makes that, were N somehow known, one can introduce it right next to h², into the framework of the ground level wave-like description (i.e. the Hamiltonian) of the entity in hand, and consequently determine the eigenvalue, and the characteristic length induced by the resulting formulation.

 

Though here, there is a peculiarity.

 

Eq.(9-2), in the simplest case of the hydrogen atom, shall (with the usual notation) be written as

E_nm_e=  n²h²   ;                                                (9) (rewritten)                   

                        (for the hydrogen atom,  is unity)

                                   

here E_n is the total energy of the n^th electronic state, R_n is the corresponding characteristic size, and n the principal quantum number.

 

In the case of the hydrogen atom,  is unity, regardless n. Thus in this case i)  is unity, at the ground state, but also ii)  remains the same, at all electronic levels.

 

Neither property holds for systems of higher complexities, though as we have shown, an equation similar to Eq.(9) can well be written for any diatomic molecule, or further any wave-like entity.  

 

Since  [of Eq.(9-2)], more generally g [of Eq.(15)] appears to be purely related to the electronic structure of the entity in hand, we expect them to remain the same, for alike electronic configurations. This occurrence holds within the frame of alike electronic states of a given molecule, as well as within the frame of alike ground states of molecules belonging to a given chemical family.

 

However, as one jumps from the ground state of a complex system, such as that of a diatomic molecule, to an excited state of this entity, it is not obvious that the electronic configuration shall stay the same; in fact, generally it will not. Take for instance the hydrogen molecule. Its excited electronic states a priori, will not bear the same electronic configuration as that of the ground state, unless the two electrons are excited in a complete symmetry. Even then, the shielding effects may not be the same.

 

 

 

 

 

This is the peculiarity we wanted to clarify; thus, as the molecule jumps from its ground state to an excited state, in general, it is not only that, h² is multiplied within the framework of the wave-like description, by the appropriate composite quantum number; but we should further represent the change that takes place in the electronic structure. That can be taken care of, by a corresponding change in the coefficient  of Eq.(9-2).

 

In fact, altering just h², and altering both h² and _, so that  is changed by the same amount, within the frame of Eq.(9), are mathematically equivalent operations; yet as discussed, physically they appear to be quite different.

 

Thereby we can conceive an excited electronic state as achieved in two steps: 1) Switching the ground state electronic configuration, into the new configuration by just changing. 2) Jumping from this configuration to the new quantum state bearing the same configuration.

 

For electronic states configured like the ground state, we will have to achieve only the second step.

 

This yields the content of our Assertion 3.

 

Assertion 3: Were the atomic or molecular wave-like object in hand, at a given electronic state, characterized by the composite quantum number N,  then the eigenvalue and the characteristic length associated with this state, becomes the output of the formulation one obtains by multiplying h² with N, in the framework of the ground state description, provided that the two states are configured similarly.

 

So the introduction of appropriate quantum numbers in Eq.(9-2), next to  (within the framework of the wave-like description), in order to take care of the excited electronic eigenstates of the molecule as complex as this may be, appears to be as standard as this is, simplest wave-like objects, such as the hydrogen atom, provided that the two states are configured similarly.

 

We can predict the solution of the new set up, through Assertion 1; it can indeed be obtained based on a reformulation of this assertion, since evidently multiplying  by a given number, and dividing the masses involved by the Hamiltonian, are mathematically identical operations. Thus we establish our Assertion 4 regarding an excited electronic level of the wave-like object in hand.

 

 

 

 

 

 

Assertion 4:   In a “real wave-like ground description” if, in the aim of expressing an excited eigenstate,  is multiplied by the composite quantum number N (which can be described as the inverse of the eigenvalue related to this eigenstate, were the ground state energy normalized to unity), then concurrently, a) the magnitude of the total ground energy E₀ associated with the given wave-like object, is decreased as much, to become E, the new eigenvalue, and b) the corresponding ground state size  stretches as much, to become R, the new size, provided that the two states are configured similarly; in mathematical words this is    

 

                        []Þ  {[], []}.              (16)

 

Note that Assertion 4 holds for any excited eigenstate (rotational, vibrational, electronic, or else).

 

This assertion, for excited states of the molecule, configured like the ground state, yields at once

                                               (17)                  (composite quantum number of the excited eigenstates,

 were  this configured like the ground state).

 

This interestingly holds no matter how complex the molecule may be.

 

Accordingly we establish our next assertion.

 

Assertion 5:             The composite quantum number to be associated with an excited eigenstate, is the mere ratio of the size the object displays at this excited state, to the size the object displays at the ground state, provided that the two states are configured similarly.

 

Assertion 5 can be checked for the electronic states of hydrogen atom. It is surprising that it holds for any object and for any excited eigenstate the object may involve.

 

What if the electronic structure of the excited state is not the same as that of the ground state?

 

The answer is fortunately not complicated. Since the coefficient  in Eq.(9) comes to multiply the mass of the electron, which happens to be the only mass taking place in the description of the electronic motion of the diatomic molecule, any change in , evidently can be represented by a corresponding hypothetical change in the mass of the electron.

 

 

 

 

 

If further, we are concomitantly to consider a quantum number N  to be associated with the excited eigenstate in question (i.e. configured in a different way than the ground state), then based on Eq.(9), this state can well be described by merely altering h²/m_e in the framework of the ground state of the molecule by N, where the subscripts “_initial” and “_final” refer respectively to the ground state and the excited electronic state in consideration.

 

The ultimate output, can be right away established via Assertions 1 and 2.

 

Assertion 6:             The ratio of the size a diatomic molecule displays at an excited state, to the size it displays at the ground state, is equal to                    N, i.e. the composite quantum number to be associated with the excited state, times a coefficient, the inverse of which quantifies how much the ground state electronic configuration is overall altered.

 

In what follows we shall solely focus on excited electronic eigenstates [since we visualize Eq.(15), for just the lowest vibrational state of an electronic eigenstate].

 

Note that the usage of Eq.(17) along Eq.(15), requires that the coefficient g is not altered as the molecule passes from its ground level to the given excited electronic state, to allow the plot of T, the largest vibrational period at the given excited electronic state, versus , where R  is the size of concern, at this eigenstate.

 

5.  THE  DISCLOSURE  OF  THE  AGED EMPIRICAL  RELATIONSHIP  

wr= Constant, AND THE COMPLETE SET OF H₂ ELECTRONIC VIBRATIONAL DATA

 

Recall that the following approximate empirical relationship, evoking very much Eq.(15), had been established for a given diatomic molecule, back in 1925, yet not unveiled so far:[9]^{,[10],[11],[12],[13]}  

 

 Empirical Constant ;                                                          (18)

 

(approximate relationship written in 1925

     for the electronic states of a given molecule)

 

here,  is the fundamental vibration frequency, i.e. the inverse of the largest vibrational period T, related to a given electronic state of the molecule, and  the corresponding internuclear distance.

 

The “Empirical Constant” is then to be determined separately, for each diatomic molecule.

 

Eq.(18) bears the same cast as that of Eq.(15) (as far as the dependency of the vibrational period on the internuclear distance is concerned); yet it does not include the quantum numbers.

 

 

 

Eq.(15), together with Assertion 5, alternatively suggests that we should look at the relationship

                                     ,                                                        (19)

               (relationship written for the largest vibrational period

                of excited electronic states of a given molecule)

 

where  is the internuclear distance at the very ground state, as usual.

 

 taking place in the above relationship, following Assertion 5, is just the composite quantum number to be associated with the electronic state taken in consideration. Yet in order to better display the structure of the interrelation between T, M₀ and r, we will not incorporate with , and keep Eq.(19) as it is, wherever this is more explanatory.

 

Eq.(19) makes that based on any molecule, regarding the electronic states bearing similar configurations, for which g, the bond looseness factor, remains about the same,  versus  should display a straight line.

 

The approximate empirical constant of Eq.(18), can now be evaluated from Eq.(19), as

                        Empirical Approximate Constant ;                           (20)

 

recall that N is the composite quantum number, i.e.  (staying indeed roughly the same, were r is not far from r₀), making up that the “constant” is question is indeed only approximately a constant, supposing anyway that the electronic states in question, are configured similarly, so that g stays practically constant, throughout.

 

This entirely discloses the mechanism behind the approximate empirical relationship [Eq.(20)], established back in 1925.

 

Thus, Eq.(20) makes that, it is not really the quantity  which is a constant for electronic states of a given molecule, configured similarly, but based on Eq.(19), more likely it is the quantity

Constant =  ;                                                      (21)

 

(written by the author, for similar

electronic states of a given molecule)

 

 

 

 

 

 

this new constant then is

Constant ;                                           (22)

 

(written by the author for similar

electronic states of a given molecule)

 

recall that  dominates the internuclear distance, at the ground state.

 

Although  stays the same for all pairs of  and r (for a given molecule), we still choose to keep it at the RHS of Eq.(21), to allow a comparison of it with the classical empirical constant of Eq.(20).

 

As an example,  versus  for H₂ molecule, is sketched in Figure 8. Thus some 23 states out of 29, for which data is available, are neatly aligned. Herein, we included , which too seems to display the same g as that of H₂ ground state; we find g0.8. The remaining 6 electronic excited states of H₂ seem to be configured differently. We call these “ambiguous states” (the previous “unambiguous” 23, being seemingly all configured more or less, like the molecule’s ground state).

 

The study of the electronic vibrational data of H₂ molecule is undertaken elsewhere.⁸

 

To analyze the remaining 6 data (out of 29), we note, out of Eq.(15) that, switching the nuclei reduced mass M₀ of alkali molecules or alkali hydrides into that of the hydrogen molecule, should virtually transpose the corresponding vibrational period, into the vibrational period of H₂ electronic state of the same electronic character; recall that switching the nuclei mass does not practically affect the electronic structure of the molecule, and accordingly we should expect that, amongst H₂ electronic states there are states, configured like the ground electronic states of alkali molecules and alkali hydrides.  

 

Therefore we anticipate that the 6 ambiguous electronic states of H₂ should be configured just like the respective ground electronic states of alkali molecules and alkali hydrides, and vice versa.

 

6. SYSTEMATIZATION OF GROUND STATES OF ALL DIATOMIC MOLECULES

 

Our approach makes that we can visualize Eq.(19) not only regarding the electronic states of a given molecule, but also regarding the ground states of molecules belonging to a given chemical family, thus exhibiting similar electronic configurations, with virtually the same g.

 

Let us elaborate on this a little.

 

 

 

Above we have rigorously proven that Eq.(15) holds for any diatomic molecule, i.e.

 

,                           (15) (rewritten)

 

 being quantum numbers induced by the Planck Constant [cf. Eq.(9)(rewritten)].

 

Within the frames of Assertions 4 and 5, regarding the electronic states of a given molecule, we have established that  turns out to be the ratio of the internuclear distance of the molecule at the given excited state, to the internuclear distance of the molecule at the ground state, provided that these states are configured alike.

 

We have further demonstrated that already the cast  holds fairly well regarding diatomic molecules belonging to a given chemical family, thus being configured similarly, so that g stays virtually the same, throughout each one of the Figures 1-7.

 

Further straightening up of these curves, requires to specify .

 

At this stage consider Figure 8, where we analyzed  spectroscopic data, and found out that the ambiguous states are configured like alkali hydrides, and .

 

This suggests that, quantum mechanically we can well describe, say the ground state of , on the basis of  an equivalent  excited state.

 

Therefore the corresponding quantum numbers , we propose to associate with  ground state, in comparison with the  ground state, following Eq.(17) and Assertion 5, becomes the mere ratio of the internuclear distance of  at its ground state, to the internuclear distance of  at its ground state, given that the  andbonds, are configured similarly.

 

Hence, we rewrite Eq.(19) (not for the excited levels of a given molecule), but for the ground states of molecules belonging to a given chemical family, and accordingly being configured alike:

 ;                                                        (23)

    (written by the author for the ground vibrational

     period of molecules belonging to a given chemical family)

 

 

 

 

here  is the ground state largest vibrational period of the ith member of the chemical family in consideration; M_0i is the reduced mass and;  is the ground state internuclear distance of this member;  is the internuclear distance of the ground state of the family’s member, chosen as the reference molecule; more precisely we pick up as the member bearing the lowest vibrational period.

 

Therefore  versus  for chemically alike molecules, should display a linear behavior, the slope of which shall furnish g, to be associated with the chemical family in consideration.

 

Thus we can now write an equation similar to Eq.(21), in regards to the ground states of molecules belonging to a given chemical family:

 

                                            ,                        (24)

 

(written by the author, for the ground states

of chemically alike molecules)

 

where  is the inverse of the ground state vibrational period of the molecule of concern.

 

Thus, the constant in question shall be expressed as

 

                                           .                                                  (25)

 

Although  is a constant within a given chemical family, we still included it, in the RHS of Eq.(24).

 

In Figures 9-15, based on experimental data,^13,[14],[15] we present  versus , for seven chemical families, for which the coefficient g, stays indeed neatly constant. The constancy of , in harmony with Eqs.(24) and (25), is quantitatively demonstrated, in (the fifth column of) Table 1-7.

 

g’s are calculated from Eq.(25) for different chemical families, and are presented in Table 8. Note that g’s vary between 0.4 and 0.01.

 

Recall that following Eqs. (24) and (25), the value of the constancy of  depends, both on g and  (the reference internuclear distance of the family of concern), which makes that the “constants” calculated in (the fifth columns of) Tables 1-7, differ.

 

Note further that, the standard deviation on the constants in question, is roughly ten percent. There seems to be two reasons for this. The first one is that chemically alike molecules, on the contrary to our assumption, are not exactly configured similarly, which may make that g is not a constant throughout. The second one is that our supposition that the RHS Eq.(17), can be used to replace the composite quantum number  in Eq.(15), even for chemically alike molecules (where we choose the molecule with the lowest vibrational period, as the reference molecule), may not be rigorous. Along this line it seems interesting to recall that, when we use the principal quantum numbers associated with the bond electrons, straight (i.e. with no quantum defects), to compose , instead of using Eq.(17), we come out with the constancy of , which happens to be not any worse than that of   [cf. Eq.(24)].^7,8

 

Above, we have predicted that the inverse of g somewhat characterizes the strength of the bond of concern [cf. the footnote we developed in relation with Eqs. (9-1) and  (9-2)]; as one can observe from Table 1, g indeed decreases as the bond becomes stronger. Thus, the higher is the number of the covalent bonds, making the overall bond of the diatomic molecule, the smaller will g be. Or the higher is the number of free electrons an atom possesses, the looser will be the bond it will make with say, an halogen, thus the higher will g be, etc. [16]

 

 

      Table 1 Checking the Validity of Eq.(23), for Alkali Molecules

 

Molecules	M0 (amu)	(cm-1 x103c)[††]			as  Referred to the Average
H2	0,50	0,24	0,74	0,62	0,29
Li2	3,50	2,89	2,67	0,40	0,15
LiNa	5,33	3,89	2,90	0,40	0,17
Na2	11,50	6,34	3,08	0,40	0,15
NaK	14,48	8,06	3,50	0,37	0,22
K2	19,49	10,80	3,92	0,37	0,22
KRb	26,83	13,2	4,07	0,36	0,24
Rb2	42,47	17,3	4,21	0,36	0,24
RbCs	52,04	20	4,42	0,35	0,27
Cs2	66,47	23,8	4,64	0,34	0,29
Average				0,40	0,22

 

 

 

 

 

 

 

 

 

Table 2 Checking the Validity of Eq.(23), for O₂ - like Molecules

 

Molecules	M0 (amu)	(cm-1 x103c)			as Referred to the Average
O2	8,00	0,64	1,21	0,15	0,17
S2	15,99	1,39	1,89	0,12	0,06
Se2	39,97	2,56	2,16	0,12	0,06
Te2	63,82	4,00	2,59	0,11	0,14
SO	10,67	0,90	1,49	0,14	0,09
Average				0,13	0,10

 

 

Table 3 Checking the Validity of Eq.(23), for N₂ - like Molecules

 

Molecules	M0 (amu)	(cm-1 x103c)			as Referred to the Average
N2	7,00	0,43	1,09	0,13	0,08
P2	15,49	1.29	1,89	0,11	0,08
PN	9,65	0,76	1,49	0,11	0,00
Average				0,12	0,05

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4 Checking the Validity of Eq.(23), for Halogens

 

Molecules	M0 (amu)	(cm-1 x104c)			as Referred to the Average
F2	11,21	9,50	1,44	1,37	0,05
Cl2	17,96	17,49	1,99	1,22	0,15
Br2	31,15	39,96	2,28	1,70	0,18
I2	46,87	63,47	2,67	1,78	0,24
BrF	15,04	15,35	1,76	1,4	0,28
ClF	12,93	12,31	1,63	1,37	0,05
ICl	26,23	27,42	2,32	1,26	0,13
Average				1,44	0,15

 

 

 Table 5 Checking the Validity of Eq.(23), for CsBr - like Molecules

 

Molecules	M0 (amu)	(cm-1 x104c)			as Referred to the Average
CsBr	52,63	49,92	3,14	1,02	0,52
CsI	71,63	64,94	3,41	1,00	0,5
NaCl	26,46	13,95	2,51	0,56	0,17
NaBr	31,98	17,86	2,64	0,60	0,09
NaI	35,15	19,45	2,90	0,54	0,19
KF	25,64	12,78	2,55	0,51	0,24
KCl	35,95	18,59	2,79	0,55	0,17
KBr	43,55	26,26	2,94	0,65	0,02
KI	47,48	29,89	3,23	0,61	0,09
RbCl	39,53	25,07	2,89	0,66	0,00
Average				0,67	0,20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 6 Checking the Validity of Eq.(23), for BF - like Molecules

 

Molecules	M0 (amu)	(cm-1 x104c)			as Referred to the Average
BF	7,26	6,72	1,26	1,44	0,69
BCl	12,06	8,38	1,72	0,88	0,03
BBr	14,77	9,66	1,88	0,80	0,06
AlCl	20,95	15,24	2,13	0,88	0,03
AlBr	26,64	20,11	2,29	0,92	0,08
InCl	31,71	26,82	2,31	1,11	0,3
InI	56,72	60,32	2,86	1,36	0,59
TlCl	35,09	29,87	2,55	1,02	0,19
TlBr	52,27	57,98	2,68	1,50	0,76
TlI	66,67	78,31	2,87	1,61	0,89
Average				1,15	0,36

 

 

Table 7 Checking the Validity of Eq.(23), for CO - like Molecules

 

Molecules	M0 (amu)	(cm-1 x104c)			as Referred to the Average
CO	4,67	6,86	1,13	2,48	0,46
CS	7,86	8,73	1,53	1,55	0,08
SiO	8,13	10,18	1,51	1,81	0,07
SiS	13,43	14,93	1,93	1,43	0,16
GeO	10,23	13,15	1,65	1,83	0,08
SnO	12,27	14,09	1,84	1,51	0,11
SnS	20,62	25,25	2,06	1,77	0,06
PbO	14,00	14,85	1,92	1,40	0,17
PbS	23,49	27,72	2,39	1,46	0,14
Average				1,69	0,15

 

 

 

 

 

 

Table 8 Bond Looseness Factors of the Chemically Alike

Diatomic Molecules

 

 

7. CONCLUSION

 

It was the author’s original idea that, owing to the end results of the special theory of relativity, as well those of the general theory of relativity, the space size, the clock mass, and the period of time to be associated with any real wave-like object, ought to be organized in just a given manner, i.e. (period of time) ~ (clock mass)(space size)²; we call this occurrence the universal matter architecture, or in short, the UMA cast.

 

In this work we were able to demonstrate this occurrence regarding the vibrational structure of diatomic molecules, to end up with the simple relationship, Eq.(15).

 

This equation should hold, generally. Thus, first of all, it should hold, regarding the electronic states of a given molecule; this interestingly led us to the empirical relationship, i.e. Eq.(18), known since 1925, but not disclosed up to our approach. We had though work out the related quantum numbers; through the approach we have developed, we could figure them out, with no difficulty; thus the product of the quantum numbers to be associated with an electronic state configured like the ground state, is the mere ratio of the internuclear distance at the excited state in consideration, to the internuclear distance at the ground state.

We further conjectured that Eq.(15) can be applied to molecules belonging to a given chemical family.

 

Indeed, the factor g, which we called “bond looseness factor”, appearing in this equation, as we have demonstrated, depends only on the electronic structure of the molecule in hand.

 

We have provided a mathematical expression for g, for those who in the future, may want to calculate it theoretically.

 

Throughout, we have determined g, based on the data, for different chemical families; thus g is a transferable quantity (i.e. once it is known for a molecule belonging to a chemical family, it can be used for all other molecules of this family).

 

The beauty of our approach, we believe, resides not only in this, but in the fact that, despite a very cumbersome quantum mechanical description to be considered for molecules, it allows us to disclose a rather simple structure behind, and grasp the related architecture thoroughly.

 

Note that Eq.(19), along Eq.(8) frames the force constant k of the molecule at the excited state of concern, as

                                          ,                                                                         (26)

 

in terms of the electron’s charge e, and the dimensionless constant f, if we cared to define f as

;                                                                (27)

recall that  is the internuclear distance at the ground state.

 

Eq. (27) is just a definition; thus, we define f in terms of the internuclear distance and the coefficient  [defined through the footnote we worked out at the level of    Eq.(9-2)].

 

Nevertheless the expression of the definition of f, can be found interesting from different points of view. First of all, the equation , one can write  from Eq.(27), is the relationship between the electron’s mass  and the orbit radius , for the hydrogen atom, based on Bohr Atom Model, where then, both f and  shall take the value of unity. Thus Eq.(27) tells us that the “clock mass” (here, the electron mass) and the “size”, owing to the electric charges, should be structured as inversely proportional to each other; the proportionality constant in question, is essentially the square of the Planck Constant. This induces the fact that, if for any reason the size is altered, in the building of a new entity, this is because the electric charges (thus the corresponding electric force), also the electron mass (thus the corresponding inertial force), are somewhat shielded. Yet the law imposing to the clock mass and the size to get structured as inversely proportional to each other, still holds.

 

Along the approach we undertook, we can further write the following relationship,[§§] involving f, for the electronic energy of the molecule of concern, as[17]^,[18],[19]

             ;                                                                            (28)

 

one can check that, f stays fairly constant for molecules belonging to a given chemical family.

 

On the other hand, Eq.(26) dimension-wise, is somewhat obvious, if one proposes to relate the force constant to the internuclear distance. This correlation was in effect proposed sometime ago, by Bratoz et al., for alkali hydrides,[20]^,[21] for which f is reported to be 2. Our estimation, based on the data¹² is, on the average, 2.6.

 

f was subsequently obtained by Salem and Ohwada[22]^,[23] which then, based on empirical presumptions, chiefly for molecules containing alkali atoms, leads to

 

                                      ,                                                                             (29)

 

where N_i and N_j, are the respective number of electrons residing outside of the complete shells of the atoms making up the diatomic molecule.

 

Note thence that, under this form f, thus g, indeed stay constant, just the way we had originally conjectured.

 

Eq.(26) yields 8 for alkali halides, whereas based on the data, and on the average, we come out with 11.1.  

 

Recall nonetheless that in order to obtain our results, we followed a totally different path, than that induced by Eq.(26). Moreover we arrived at our result, primarily regarding the electronic states of a given molecule. The literature we reviewed does not coop at all with such an aspect.    

 

Note further that recent trials, on the “problem of transferable spectroscopic constants”, despite satisfactory results they may furnish, are far from displaying how the fundamental quantities of mass, space and time (i.e. clock mass, clock size and period of time of the clock motion), are structured in interrelation with each other, in the architecture of molecules,[24] in fact just the way Eq.(19) reveals.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Author:            Tolga Yarman, Ph. D.

 

Title:                Professor

 

e-mail:            tyarman@isikun.edu.tr

 

Affiliation:       Isik University, Physics Department

 

web-site:        www.isikun.edu.tr , http://phys.isikun.edu.tr

 

Address:        Isik University, Maslak, Istanbul, Turkey.

 

Signature:

 

Keywords:      systematic diatomic molecules, quantum numbers of diatomic molecules

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ACKNOWLEDGEMENT

 

The author would like extend his profound gratitude to Dear Friend, Professor V. Rozanov. He would like to further thank to Drs. N. Veziroðlu, O. Sinanoðlu, E. Hasanov, C. Marchal, ª. Koçak, and V. Altýn, and to his Dear Brothers Dr. S. B. Yarman and Dr. F. Yarman, for very many hours of discussions, which helped tremendously to improve the work presented herein.

 

The author would further like to thank to Dear Research Asistant Fatih Özaydýn, who has kindly helped the typing of the manuscript and drawn the figures.

 

 

REFERENCES