A PHYSICAL CONSTRAINT METHOD OF ESTIMATING THE MIXING PARAMETERS FOR XY4 MOLECULES AND ANALYSIS OF TEN ELLIPSES

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Volume 2013, Issue No.4, April 22 2013, Time: 8h55m A. M.

A PHYSICAL CONSTRAINT METHOD OF ESTIMATING THE MIXING PARAMETERS FOR XY₄ MOLECULES AND ANALYSIS OF TEN ELLIPSES 
by

Professor Dr. Kotcherlakota Lakshmi Narayana

{Retd. Prof. of Physics, SU, Kolhapur}, 17-11-10, Narasimha Ashram, 
Official Colony, Maharanipeta.P.O, Visakhapatnam-530002

Mobile No:  9491902867 & 9594717723

 ABSTRACT

                                     K. L. Narayana’s (1973) method has suggested a 

possible estimate of the XY₄ molecules mixing parameter under 

the physical constraint that the interaction force constant frφ 


between a bond and an angle adjacent is equal to the 

interaction force constant frφ’ between a bond and an angle 

diagonally opposed. A complete analysis of the ellipses has been 

performed afresh and the entire data of about ten ellipses 

listed. 

         A very interesting aspect of Force Constants and the area of 

ellipses is conjectured.

           Below I present a completely new table of  data analysis of the force constants and the ellipses, giving several details of utmost physical significance.

DETAILS

               The constraint is equivalent to the mathematical condition that the force constants

 F₄₅= F₅₄= F₆₇=F₇₆=0………………………………....Eq.1

The method described below results in a quadratic expression for the mixing parameter. 
The force constant matrix of Wilson’s 

Fw= (L^-1) ^T Λ L ^-1…………………………………...…Eq.2

where

                         L= BA………………………………………………..............Eq.3

                         L= BA

                         = [cB₄₄/C              B₄₄/C                 0                        0;

                             (cB₅₄+B₅₅)/C   (B₅₄-cB₅₅)/C    0                        0;

                                0                      0                         c B₆₆/C              B₆₆/C;

                                0                      0                       (cB₇₆+ B₇₇)/C   (B₇₆- cB₇₇)/C]…..Eq.4

in which for XY₄  type molecules it may easily be verified 

B₄₄= B₆₆;

    B₄₅=B₆₇=0;

B₅₄=B₇₆;

               and B₆₆= B₇₇;…………Eq.5

By finding the inverse of this and substituting in the expression for Fw given above given in eq.2 we get

F₄₅= λ₄ B₄₄ (B₅₄- cB₅₅) + λ₅ c B₄₄ (cB₅₄+B₅₅)   ………………….Eq.6

Hence

B₅₄ (λ₅\λ₄) c^2 + B₅₅ (λ₅\λ₄ -1) c+ B₅₄=0 ……………………..……Eq.7

For example in the case of PtCl4 the values of B=matrix elements are

   B₄₄= -0.1971;

B₅₅=-0.269;

   B₅₄=0.07363;

and

λ₄= 0.0588003(mdyne/^A0)/amu

                                   λ₅= 0.0201534(mdyne/A⁰)/amu………………..Eq.8

using these we obtain the two possible mixing parameters values as

c= -6.565;

                           and c= -0.4396;……………...................….Eq.9

Adopting a value of c as c=-6.0 the molecular constants for PtCl₄ are evaluated and they are listed below,

F₄₄= 1.406 mdyne/A⁰

     F₄₅= -0.008576 mdyne/A⁰

                                       and F₅₅= 0.2861 mdyne/A⁰………………….Eq.10.

                       The low values of F₄₅ (which should have been actually zero as per the quadratic equation given above) are because the used mixing parameter value is -6.0 while from the quadratic equation it is -6.565. From a comparison of these, with those obtained by Sabitani et al. [19] with the assumption of F₄₅=0, it is to be concluded therefore that the small change of this amount in the value of F₄₅ changes the other force constant values considerably in the second decimal. The Green’s function method of estimating the force constants has been claimed to significant in almost two figures by Wolfram et al. [21]. The calculated mean square amplitudes, potential energy distribution and the L-matrix elements are listed below.

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A NEW TABLE OF FORCE CONSTANTS AND AREA OF ELLIPSES

                  Below I present a completely new table of data analysis of the force constants and the ellipses, giving several details of utmost physical significance.

			FORCE CONSTANTS AND AREA OF ELLIPSES
Molecule	Sample	Major			a*cos	π*a*b (Cm^2)	SCALES USED
PtCl4		a Cm	Degree	Cos value	b*cos	area ellipse	axis x	axis y
Fig.4	F44 Vs F45	7.5	25	0.906308	6.797308403	103.6725576	0.005/mm	.01/mm
minor	b Cm	4.4	25	0.906308	3.987754263
Fig.4	F55 Vs F45	7.7	30	0.866025	6.668395609	117.3227776	.005/mm	0.005/mm
minor	b Cm	4.85	30	0.866025	4.200223208
Fig.5	∑44 Vs ∑45	7	-27.5	0.887011	6.209075832	98.96016859	.0001/mm	.0001/mm
minor	b Cm	4.5	-27.5	0.887011	3.991548749
Fig.6	∑55 Vs ∑45	7	-20	0.939693	6.577848346	92.36282402	.0001/mm	.0002/mm
minor	b Cm	4.2	-20	0.939693	3.946709007
AuCl4
Fig.7	F44 Vs F45	5.8	22	0.927184	5.377666356	69.24070209	.01/mm	.02/mm
minor	b Cm	3.8	22	0.927184	3.523298647
Fig.8	F55 Vs F45	5.5	29	0.87462	4.810408389	62.20353454	.01/mm	.01/mm
minor	b Cm	3.6	29	0.87462	3.148630946
Fig.9	∑44 Vs ∑45	6.6	-25	0.906308	5.981631394	93.30530181	.0001/mm	.0001/mm
minor	b Cm	4.5	-25	0.906308	4.078385042
Fig.10	∑55 Vs ∑45	7	-25	0.906308	6.344154509	92.36282402	.0001/mm	.0002/mm
minor	b Cm	4.2	-25	0.906308	3.806492706
PdCl4
Fig.11	F44 Vs F45	8.9	28	0.882948	7.858233576	150.9849429	.01/mm	.005/mm
minor	b Cm	5.4	28	0.882948	4.767917001
Fig.11	F55 Vs F45	8.8	29	0.87462	7.696653423	147.9061821	.005/mm	.005/mm
minor	b Cm	5.35	29	0.87462	4.679215433
Fig.12	∑44 Vs ∑45	7.5	-15	0.965926	7.244443697	87.17919614	.0001/mm	.0002/mm
minor	b Cm	3.7	-15	0.965926	3.573925557
Fig.13	∑55 Vs ∑45	8.5	-17	0.956305	8.128590426	122.8362728	.0001/mm	.00025/mm
minor	b Cm	4.6	-17	0.956305	4.399001877
ICl4
Fig.14	F44 Vs F45	9.2	70	0.34202	3.146585319	132.9522011	.005/mm	.005/mm
minor	b Cm	4.6	70	0.34202	1.573292659
Fig.14	F55 Vs F45	6	30	0.866025	5.196152423	58.43362336	.005/mm	.005/mm
minor	b Cm	3.1	30	0.866025	2.684678752
Fig.15	∑44 Vs ∑45	8	-25	0.906308	7.250462296	113.0973355	.0001/mm	.0001/mm
minor	b Cm	4.5	-25	0.906308	4.078385042
Fig.16	∑55 Vs ∑45	7.7	-24	0.913545	7.034300024	101.5991064	.0001/mm	.00025/mm
minor	b Cm	4.2	-24	0.913545	3.836890922
XeF4
Fig.17	F44 Vs F45	5	20	0.939693	4.698463104	50.26548246	.02/mm	.02/mm
minor	b Cm	3.2	20	0.939693	3.007016387
Fig.18	F55 Vs F45	7	68	0.374607	2.622246154	103.3583983	.02/mm	.02/mm
minor	b Cm	4.7	68	0.374607	1.760650989
Fig.19	∑44 Vs ∑45	5.6	-14	0.970296	5.433656067	59.81592412	.0002/mm	.0002/mm
minor	b Cm	3.4	-14	0.970296	3.299005469
Fig.20	∑55 Vs ∑45	5.8	-9.8	0.985408	5.715365811	58.30795965	.0005/mm	.0002/mm
minor	b Cm	3.2	-9.8	0.985408	3.153305275
AuBr4
Fig.21	F44 Vs F45	9	62	0.469472	4.225244065	144.1991028	.01/mm	.01/mm
minor	b Cm	5.1	62	0.469472	2.39430497
Fig.21	F55 Vs F45	9.2	60	0.5	4.6	150.2937925	.01/mm	.005/mm
minor	b Cm	5.2	60	0.5	2.6
Fig.22	∑44 Vs ∑45	10.5	-21	0.93358	9.802594478	181.4269757	.0004/mm	.00005/mm
minor	b Cm	5.5	-21	0.93358	5.134692346
Fig.23	∑55 Vs ∑45	10.2	-15	0.965926	9.852443428	131.3814048	.00015/mm	.0004/mm
minor	b Cm	4.1	-15	0.965926	3.960295888
Pt(CN)4
Fig.24	F44 Vs F45	5.4	0	1	5.4	64.46548125	.005/mm	.005/mm
minor	b Cm	3.8	0	1	3.8
Fig.25	F55 Vs F45	3.6	16	0.961262	3.460542105	29.40530724	.005/mm	.005/mm
minor	b Cm	2.6	16	0.961262	2.499280409
Fig.26	∑44 Vs ∑45	5.7	0	1	5.7	68.04689688	.0005/mm	.0005/mm
minor	b Cm	3.8	0	1	3.8
Fig.27	∑55 Vs ∑45	6	-4	0.997564	5.985384302	75.39822369	.0005/mm	.0005/mm
minor	b Cm	4	-4	0.997564	3.990256201
Pd(CN)4
Fig. 28	F44 Vs F45	7.9	81	0.156434	1.235832274	143.9477754	.005/mm	.005/mm
minor	b Cm	5.8	81	0.156434	0.907319897
Fig.29	F55 Vs F45	6	13	0.97437	5.846220389	67.85840132	.005/mm	.005/mm
minor	b Cm	3.6	13	0.97437	3.507732233
Fig.30	∑44 Vs ∑45	6	-8	0.990268	5.941608412	75.39822369	.0005/mm	.0005/mm
minor	b Cm	4	-8	0.990268	3.961072275
Fig.31	∑55 Vs ∑45	5.8	-12.5	0.976296	5.662516841	76.52919704	.0005/mm	.001/mm
minor	b Cm	4.2	-12.5	0.976296	4.10044323
Ni(CN)4
Fig.32	F44 Vs F45	6.3	22	0.927184	5.841258284	81.14733824	.005/mm	.05/mm
minor	b Cm	4.1	22	0.927184	3.801453804
Fig.33	F55 Vs F45	6.3	19	0.945519	5.956767026	85.10574499	.005/mm	.005/mm
minor	b Cm	4.3	19	0.945519	4.065729875
Fig.34	∑44 Vs ∑45	6.65	-16	0.961262	6.392390278	90.87842149	.0005/mm	.0005/mm
minor	b Cm	4.35	-16	0.961262	4.181488377
Fig.35	∑55 Vs ∑45	8.5	-18	0.951057	8.083980389	120.165919	.0005/mm	.0005/mm
minor	b Cm	4.5	-18	0.951057	4.279754323
Au(CN)4
Fig.36	F44 Vs F45	7.3	84	0.104528	0.763057782	114.6681319	.005/mm	.005/mm
minor	b Cm	5	84	0.104528	0.522642316
Fig.37	F55 Vs F45	5	-6	0.994522	4.972609477	58.11946409	.005/mm	.005/mm
minor	b Cm	3.7	-6	0.994522	3.679731013
Fig.38	∑44 Vs ∑45	5.6	0	1	5.6	70.37167544	.0005/mm	.0005/mm
minor	b Cm	4	0	1	4
Fig.39	∑55 Vs ∑45	5.4	-17	0.956305	5.164045682	67.85840132	.0005/mm	.001/mm
minor	b Cm	4	-17	0.956305	3.825219024

The minor and major refer to the Ellipse axises.

           Force Constant Ellipses for the set of ten molecules given below:

 Fig.4

Fig.5

Fig.6

Fig.7 

Fig.8

Fig.9

Fig.10

Fig.11

Fig.12

Fig.13

Fig.14

Fig.15

Fig.16

Fig.17

Fig.18

Fig.19

Fig. 20

Fig..21

Fig.22

Fig.23

Fig.24

Fig.25

Fig.26

Fig.27

Fig.29

Fig.28

Fig.30

Fig.31

Fig.32

Fig.33

Fig.34

Fig.35

 Fig.36

 Fig.37

Fig.38

Fig.39

REFERENCES

                  

ACKNOWLEDGMENT

               I have been greatly benefited from the Internationally Famous laboratories of Late Prof. K. R. Rao, D.Sc.(Madras) D.Sc.(London) to whom I am deeply indebted. The specific research work and analysis of the data I have learned from his laboratories. Some contents of this paper forms, a part of my student Ph. D thesis of Shivaji University, Kolhapur where I was employed as a Physicist from June 1, 1966 to July 31, 2000.