Molecular van der Waals Space and Topological Indices from the Distance Matrix

Abstract

A comparative study of 36 molecular descriptors derived from the topological distance matrix and van der Waals space is carried out within this paper. They are partitioned into 16 generalized topological distance matrix indices, 11 topological distance indices known in the literature (seven obtained from eigenvalues/eigenvectors of distance matrix), and 9 van der Waals molecular descriptors. The generalized topological distance indices, ^kδ_λ (λ = 1 – 3, k = 1 – 4), are introduced in this work on the basis of reciprocical distance matrix. Intercorrelation analysis reveals that topological distance indices mostly contain the same type of information, while van der Waals indices can be bound to the shape or the size of molecules. Furthermore, we found that topological distance indices are good for describing molecular size, and they may be viewed as bulk parameters. The most accurate QSPR models for predicting boiling point of alkanes are based on some of the generalized, eigenvalues/eigenvectors topological distance indices and the van der Waals descriptors of molecular size.

Introduction

The most important problem in QSPR and QSAR analysis is to convert chemical structure into mathematical molecular descriptors that are relevant to the physical, chemical or biological properties [1]. Molecular structure is one of the basic concepts of chemistry, since properties and chemical and biological behaviors of molecules are determined by it. One can distinguish three levels for quantifying molecular structure: topological (based on atomic connectivity) [2], metric (bond length, valence and torsion angles) [3] and electronic (quantum-mechanical evaluation of detailed dynamics of electrons and nuclei) [4]. Within many congener series of chemical compounds the variations of molecular geometry (as measured by van der Waals descriptors), and electronic structure are small [5,6]. Consequently, one can consider that many of molecular properties are conditioned only by topology of molecules and quantify the structural information contained in their molecular graphs by means of so-called topological indices (TIs). These are numerical quantities based on various invariants or characteristics of molecular graph. Among them, more detailed topological information is provided by the topological distance matrix D, whose entries d_ij represent topological distances between vertices i and j, that is the number of edges (bonds) along the shortest path between these vertices (atoms). Therefore, many TIs used in QSPR and QSAR studies have been developed on the basis of D.

From their definitions, one may admit many TIs derived from D may code two structural steric factors, namely the size and shape of the molecule [7]. Although TIs do not have a precise physical meaning, they are measures for topological shape, i.e. the degree of branching or cyclicity and they correlate well with molecular volume or surface [1]. However, extensive studies on this topic do not yet exist.

On the other hand, the idea that the molecular van der Waals (vdW) space is responsible for molecular properties affords an adequate reason for introducing vdW molecular descriptors (vdWMDs) with a clear physical meaning [3,5]. They were frequently used as molecular descriptors by themselves [3,6,8] or as a starting point for deriving other parameters, e.g. lipophilicity/ hydrophilicity [9], surface tension parameters [10], Weighted Holistic Invariant Molecular (WHIM) descriptors [11] and so on.

In this paper we present our efforts to develop some topological distance indices (TDIs) [5,12,13,14,15,16,17] and vdWMDs [3,5,6,18,19] and investigate if there exists a linear relationship between these two groups of structural parameters, situated at the first and second level of molecular structural information, respectively. One type of TDIs, the generalized (global) topological distance indices (GTDIs), denoted by ^kδ_λ, λ=0,1,2,3 and k=1,2,3,4,…, is generalized here on the basis of reciprocal distances from a molecular graph Γ (reciprocal distance matrix [20]). The other type was developed with the aid of real number local vertex invariants (LOVIs) based on the graph eigenvalues [12,13,14]. Eigenvectors corresponding to the largest negative eigenvalue of the distance matrix, D, can serve as LOVIs. Various TDIs have been obtained from these LOVIs by various operations (simple summation, or application of Randić-type formulas) [12]. All TDIs presented here were tested in correlations against boiling points of alkanes, with satisfactory results for some of them, also reported in this work. It must be mentioned that Trinajstić et al. [21] compared five TDIs and five topographical (3D) distance indices in order to answer the questions as to what extent the distance indices are intercorrelated and how they perform in a given QSAR for the boiling points of the first 150 alkanes with 2-10 carbon atoms.

Among calculated vdWMDs, [5] we selected here as molecular structural descriptors only the vdW volume V^W and surface S^W, V^W/S^W, vdW volume of molecule considered as ellipsoids V^WE, semi-axes of the ellipsoid (a,b,c) which embeds a given molecule (viewed as a collection of atomic spheres distributed in 3D-space, each atomic sphere having a radius equal with its vdW radius) and two globularity measures [3,5,6,12].

The results obtained by correlation analysis of all the above described molecular structural descriptors and a QSPR study of boiling temperatures of the first alkanes with 2-9 carbon atoms are also reported. They permit some insights about the physical meaning of the investigated TDIs.

Description of Selected Topological Distance Indices

The distance matrix D(Γ) = {d_ij} of a graph Γ is an important graph-invariant. Its entries d_ij, called distances, are equal to the number of edges connecting the vertices i and j on the shortest path between them. Thus all d_ij are integers, and d_ij =1 for nearest neighbors; by definition, d_ii = 0. Therefore, the distance matrix D = D(Γ) of a labeled connected graph Γ is a real symmetric matrix NxN whose elements d_ij are defined as [21,22]:

(1)

where l_ij is the topological length of the shortest path, i. e. the minimum number of edges between the vertices i and j in Γ. The length of the shortest path l_ij is also called [22] the distance between the vertices i and j in Γ, hence the name “distance matrix” for D.

Many TDIs have been developed on the basis of D. We selected some of these for the present study, in which we analyze the relationship between TDIs and molecular vdW space. Among the TDIs that can be derived from D, the most popular investigated and applied is the Wiener number [23]. Besides the Wiener number [24,25] we will briefly present the following TDIs used in our analysis: the polarity number [24,25,26], the Platt index [26], the Balaban J index [27,28], and TDIs based on graph eigenvalues and eigenvectors [12,13,14]. We also generalize here the TDIs derived [5,15,16,17] from reciprocal distance matrix [20,21], denoted by ^kδ_λ.

(a) Wiener index

The Wiener index, W, [24,25] was defined as the sum of the number of bonds separating all pairs of atoms in an acyclic molecule. It is easily to shown that this index equal to the half-sum of the off-diagonal elements of D [29]:

(2)

where N is the total number of vertices (atoms) in Γ.

(b) Polarity number

Wiener has also introduced the so-called polarity number, P. P is the number of pairs of vertices separated by three edges, that is half of the number of distances of length three:

(3)

In relation (3) N represents the total number of vertices in Γ.

The ½ factor before the sums in (3) compensates for the fact that the three edges between the vertices i and j in Γ are accounted for two times (both ways). W and P have been applied to correlations with boiling points, heat of formation and vaporization and other physical properties of alkanes [24,25,26].

(c) Platt index

Platt (nearest-neighbor edges) index F is calculated by summing for each edge the number of its adjacent edges [26]:

(4)

(d) Balaban index

Balaban [27,28] has proposed a topological index, which can be described as the average distance sum connectivity. The Balaban topological index J of a molecular graph Γ is defined as [27]:

(5)

where m is the number of edges in Γ, μ is the cyclomatic number, and the vertices i and j are adjacent.

The average distance sum for a vertex k in Γ represents the sum of all entries of the k^th row or column in the distance matrix, D [27]:

(6)

The cyclomatic number μ = μ(Γ), i.e. the number of cycles in Γ, is given by [28]

μ = m – n + 1

(7)

where N is the number of vertices in Γ. Relation (7) is the known Euler equation connecting the number of vertices (N), edges (m) and cycles (μ) in a planar graph. Average distance sums were used in relation (5) instead of distance sums because distance sums increase approximately parallel with m for the same type of branching. The ciclomatic number μ, defined in (7), was introduced in the definition of J because the presence of cycles markedly reduces the distance sums [7].

(e) Graph eigenvalues or eigenvector –based indices

Lowest and highest eigenvalues and corresponding eigenvectors of matrices A and D have also been used as topological indices and local vertex invariants (LOVIs) [12,13,14,30]. We present here only TDIs derived by us [12,13,14] from D of all alkanes with 2-9 carbon atoms. From the largest negative eigenvalues of D, denoted by E(D), and corresponding eigenvectors we introduced the following TDIs [13]:

VAD1 = - E(A)

(8)

(9)

(10)

where e_i are the elements (LOVIs) of the first eigenvector derived from E(D) and N is the number of carbon atoms.

Two kinds of normalizations against the number N of carbon atoms of the alkane were carried out. Each of these led to a type of TDIs, denoted below by VxDk, distinguished by the final number k = 2 or k = 3:

(11)

(12)

where x = A, E.

Up to eight carbon atoms no degeneracy was found in the TDIs values as estimated by relations (8)–(12). However, for nine carbon atoms, just one pair of isomers for VED-type indices was found to have degenerate values [13].

The VxDk (with x=A and E, and k=1,3) and VRD indices were calculated here for the first alkanes with 2-9 carbon atoms by means of our IRS [31] computer package. The values were compared with those obtained with the aid of DRAGON [32]. The W, P, F, J, VADk and VEDk (k=1,3), VRD indices for 72 alkanes with N=1–9 carbon atoms are given in Table 1a.

Table 1a. Topological Distance Indices and Boiling Points of the First 72 Alkanes

Alkane	BP	W	P	F	J	VAD1	VAD2	VAD3	VED1	VED2	VED3	VRD
C2	-88.5	1	0	0	1.0000	1.0000	0.5000	-1.6094	1.4142	0.7071	-1.2629	1.4142
C3	-44.5	4	0	2	1.6330	2.7321	0.9107	-0.1989	1.7156	0.5719	-0.6642	3.7224
C4	-0.5	10	1	4	1.9747	5.1623	1.2906	0.7251	1.9742	0.4935	-0.2361	6.5255
2-M-C3	-10.5	9	0	6	2.3238	4.6458	1.1614	0.6197	1.9723	0.4931	-0.2371	6.9009
C5	36.5	20	2	6	2.1906	8.2882	1.6576	1.4217	2.2036	0.4407	0.0970	9.7395
2M-C4	27.9	18	2	8	2.5395	7.4593	1.4919	1.3163	2.2020	0.4404	0.0962	10.1583
22MM-C3	9.5	16	0	12	3.0237	6.6056	1.3211	1.1948	2.2040	0.4408	0.0971	10.7414
C6	68.7	35	3	8	2.3391	12.1093	2.0182	1.9832	2.4118	0.4020	0.3696	13.3165
3M-C5	63.2	31	4	10	2.7542	10.7424	1.7904	1.8634	2.4085	0.4014	0.3682	13.8800
2M-C5	60.2	32	3	10	2.6272	11.0588	1.8431	1.8924	2.4117	0.4020	0.3695	13.6798
23MM-C4	58.1	29	4	12	2.9935	10.0000	1.6667	1.7918	2.4121	0.4020	0.3697	14.1487
22MM-C4	49.7	28	3	14	3.1685	9.6702	1.6117	1.7582	2.4111	0.4019	0.3693	14.4073
C7	98.4	56	4	10	2.4475	16.6254	2.3751	2.4543	2.6036	0.3720	0.6002	17.2230
3E-C5	93.5	48	6	12	2.9923	14.8636	2.1234	2.3422	2.6009	0.3716	0.5992	17.7855
3M-C6	91.8	50	5	12	2.8318	14.2970	2.0424	2.3034	2.5975	0.3711	0.5979	18.0592
2M-C6	90.0	52	4	12	2.6783	13.0698	1.8671	2.2136	2.6005	0.3715	0.5990	18.5519
23MM-C5	89.8	46	6	14	3.1442	15.4048	2.2007	2.3780	2.6050	0.3721	0.6008	17.5136
33MM-C5	86.0	44	6	16	3.3604	14.1760	2.0251	2.2949	2.6067	0.3724	0.6014	17.8657
223MMM-C4	80.9	42	6	18	3.5412	13.6346	1.9478	2.2559	2.6027	0.3718	0.5999	18.1940
24-MMC5	80.5	48	4	14	2.9532	13.6353	1.9479	2.2560	2.6038	0.3720	0.6003	18.2007
22MM-C5	79.2	46	4	16	3.1545	12.3945	1.7706	2.1606	2.6066	0.3724	0.6014	15.8479
C8	125.8	84	5	12	2.5301	21.8364	2.7295	2.8604	2.7824	0.3478	0.8002	21.4335
3E-C6	118.9	72	7	14	3.0744	19.5420	2.4428	2.7494	2.7787	0.3473	0.7989	22.0645
3M-C7	118.8	76	6	14	2.8621	19.7628	2.4704	2.7607	2.7810	0.3476	0.7997	21.9365
34MM-C6	118.7	68	8	16	3.2925	18.7788	2.3474	2.7096	2.7762	0.3470	0.7979	22.3387
3E-3M-C5	118.2	64	9	18	3.5832	16.6705	2.0838	2.5905	2.7768	0.3471	0.7982	23.1188
4M-C7	117.7	75	6	14	2.9196	17.4187	2.1773	2.6344	2.7789	0.3474	0.7989	22.7102
2M-C7	117.6	79	5	14	2.7158	17.6759	2.2095	2.6491	2.7799	0.3475	0.7993	22.5967
3E-2M-C5	115.6	67	8	16	3.3549	17.4427	2.1803	2.6358	2.7789	0.3474	0.7989	22.7488
23MM-C6	115.3	70	7	16	3.1708	20.4792	2.5599	2.7963	2.7849	0.3481	0.8011	21.6556
233MMM-C5	114.6	62	9	20	3.7083	19.1115	2.3889	2.7272	2.7878	0.3485	0.8021	21.9131
234MMM-C5	113.4	65	8	18	3.4642	18.3964	2.2996	2.6890	2.7838	0.3480	0.8007	22.2412
33MM-C6	112.0	67	7	18	3.3734	18.1815	2.2727	2.6773	2.7815	0.3477	0.7999	22.3829
223MMM-C5	110.5	63	8	20	3.6233	16.8079	2.1010	2.5987	2.7851	0.3481	0.8011	22.7627
24MM-C6	109.4	71	6	16	3.0988	16.0683	2.0085	2.5537	2.7826	0.3478	0.8002	23.1970
25MM-C6	108.4	74	5	16	2.9278	18.4133	2.3017	2.6899	2.7843	0.3480	0.8009	22.2507
22MM-C6	107.0	71	5	18	3.1118	17.0338	2.1292	2.6121	2.7878	0.3485	0.8021	22.6056
2233MMMM-C4	106.0	58	9	24	4.0204	16.3152	2.0394	2.5690	2.7838	0.3480	0.8007	23.0345
224MMM-C5	99.3	66	5	20	3.3889	14.9373	1.8672	2.4807	2.7892	0.3487	0.8026	23.5670
C9	150.6	120	6	14	2.5951	27.7422	3.0825	3.2176	2.9505	0.3278	0.9766	25.9281
33EE-C5	146.2	88	12	20	3.8247	25.0208	2.7801	3.1143	2.9471	0.3275	0.9755	26.5488
3E-C7	143.0	104	8	16	3.0922	23.6799	2.6311	3.0593	2.9429	0.3270	0.9740	26.9873
3M-C8	143.0	110	7	16	2.8766	21.7527	2.4170	2.9744	2.9458	0.3273	0.9750	27.4391
4M-C8	142.5	108	7	16	2.9548	22.6204	2.5134	3.0135	2.9491	0.3277	0.9761	27.0595
2M-C8	142.5	114	6	16	2.7467	22.2705	2.4745	2.9979	2.9448	0.3272	0.9747	27.3590
3E-23MM-C5	141.6	86	12	22	3.9192	25.4119	2.8236	3.1299	2.9505	0.3278	0.9766	26.3543
2334MMMM-C5	141.5	84	12	24	4.0137	24.0988	2.6776	3.0768	2.9457	0.3273	0.9750	26.7923
4E-C7	141.2	102	8	16	3.1753	21.3349	2.3705	2.9550	2.9439	0.3271	0.9744	27.6869
3E-3M-C6	140.6	92	10	20	3.6174	22.2198	2.4689	2.9956	2.9461	0.3273	0.9751	27.2876
23MM-C7	140.5	102	8	18	3.1553	20.7438	2.3049	2.9269	2.9501	0.3278	0.9765	27.6329
334MMM-C6	140.5	88	11	22	3.8024	19.8563	2.2063	2.8832	2.9481	0.3276	0.9758	28.0907
2233MMMM-C5	140.3	82	12	26	4.1447	23.0687	2.5632	3.0331	2.9507	0.3279	0.9767	26.8886
34MM-C7	140.1	98	9	18	3.3248	22.6789	2.5199	3.0161	2.9473	0.3275	0.9755	27.1235
234MMM-C6	139.0	92	10	20	3.5758	22.6772	2.5197	3.0160	2.9482	0.3276	0.9759	27.1184
233MMM-C6	137.7	90	10	22	3.7021	20.3172	2.2575	2.9061	2.9490	0.3277	0.9761	27.8661
33MM-C7	137.3	98	8	20	3.3301	26.2722	2.9191	3.1632	2.9537	0.3282	0.9777	26.0911
3E-24MM-C5	136.7	90	10	20	3.6776	24.7896	2.7544	3.1051	2.9575	0.3286	0.9790	26.2728
35MM-C7	136.0	100	8	18	3.2230	23.9292	2.6588	3.0697	2.9542	0.3282	0.9779	26.5647
25MM-C7	136.0	104	7	18	3.0608	23.5441	2.6160	3.0535	2.9506	0.3278	0.9767	26.7803
26MM-C7	135.2	108	6	18	2.9147	21.1839	2.3538	2.9479	2.9525	0.3281	0.9773	27.4273
44MM-C7	135.2	96	8	20	3.4311	23.5541	2.6171	3.0539	2.9507	0.3279	0.9767	26.7833
4E-2M-C6	133.8	98	8	18	3.3074	22.0627	2.4514	2.9885	2.9549	0.3283	0.9781	27.0476
3E-22MM-C5	133.8	88	10	22	3.7929	21.1970	2.3552	2.9485	2.9515	0.3279	0.9770	27.4362
24MM-C7	133.5	102	7	18	3.1513	20.7945	2.3105	2.9293	2.9490	0.3277	0.9761	27.7030
2234MMMM-C5	133.0	86	10	24	3.8776	19.3005	2.1445	2.8548	2.9542	0.3283	0.9779	28.1080
22MM-C7	132.7	104	6	20	3.0730	23.9635	2.6626	3.0712	2.9540	0.3282	0.9778	26.5871
223MMM-C6	131.7	92	9	22	3.5887	22.4662	2.4962	3.0067	2.9585	0.3287	0.9793	26.8263
235MMM-C6	131.3	96	8	20	3.3766	21.6063	2.4007	2.9676	2.9548	0.3283	0.9781	27.1920
244MMM-C6	126.5	92	8	22	3.5768	20.1263	2.2363	2.8967	2.9602	0.3289	0.9799	27.5398
224MMM-C6	126.5	94	7	22	3.4673	21.2250	2.3583	2.9498	2.9512	0.3279	0.9769	27.4608
225MMM-C6	124.0	98	6	22	3.2807	19.7257	2.1917	2.8766	2.9565	0.3285	0.9786	27.8260
2244MMMM-C5	122.7	88	6	26	3.7464	18.8440	2.0938	2.8308	2.9541	0.3282	0.9779	28.3248

Table 1a. Topological Distance Indices and Boiling Points of the First 72 Alkanes

Alkane	BP	W	P	F	J	VAD1	VAD2	VAD3	VED1	VED2	VED3	VRD
C2	-88.5	1	0	0	1.0000	1.0000	0.5000	-1.6094	1.4142	0.7071	-1.2629	1.4142
C3	-44.5	4	0	2	1.6330	2.7321	0.9107	-0.1989	1.7156	0.5719	-0.6642	3.7224
C4	-0.5	10	1	4	1.9747	5.1623	1.2906	0.7251	1.9742	0.4935	-0.2361	6.5255
2-M-C3	-10.5	9	0	6	2.3238	4.6458	1.1614	0.6197	1.9723	0.4931	-0.2371	6.9009
C5	36.5	20	2	6	2.1906	8.2882	1.6576	1.4217	2.2036	0.4407	0.0970	9.7395
2M-C4	27.9	18	2	8	2.5395	7.4593	1.4919	1.3163	2.2020	0.4404	0.0962	10.1583
22MM-C3	9.5	16	0	12	3.0237	6.6056	1.3211	1.1948	2.2040	0.4408	0.0971	10.7414
C6	68.7	35	3	8	2.3391	12.1093	2.0182	1.9832	2.4118	0.4020	0.3696	13.3165
3M-C5	63.2	31	4	10	2.7542	10.7424	1.7904	1.8634	2.4085	0.4014	0.3682	13.8800
2M-C5	60.2	32	3	10	2.6272	11.0588	1.8431	1.8924	2.4117	0.4020	0.3695	13.6798
23MM-C4	58.1	29	4	12	2.9935	10.0000	1.6667	1.7918	2.4121	0.4020	0.3697	14.1487
22MM-C4	49.7	28	3	14	3.1685	9.6702	1.6117	1.7582	2.4111	0.4019	0.3693	14.4073
C7	98.4	56	4	10	2.4475	16.6254	2.3751	2.4543	2.6036	0.3720	0.6002	17.2230
3E-C5	93.5	48	6	12	2.9923	14.8636	2.1234	2.3422	2.6009	0.3716	0.5992	17.7855
3M-C6	91.8	50	5	12	2.8318	14.2970	2.0424	2.3034	2.5975	0.3711	0.5979	18.0592
2M-C6	90.0	52	4	12	2.6783	13.0698	1.8671	2.2136	2.6005	0.3715	0.5990	18.5519
23MM-C5	89.8	46	6	14	3.1442	15.4048	2.2007	2.3780	2.6050	0.3721	0.6008	17.5136
33MM-C5	86.0	44	6	16	3.3604	14.1760	2.0251	2.2949	2.6067	0.3724	0.6014	17.8657
223MMM-C4	80.9	42	6	18	3.5412	13.6346	1.9478	2.2559	2.6027	0.3718	0.5999	18.1940
24-MMC5	80.5	48	4	14	2.9532	13.6353	1.9479	2.2560	2.6038	0.3720	0.6003	18.2007
22MM-C5	79.2	46	4	16	3.1545	12.3945	1.7706	2.1606	2.6066	0.3724	0.6014	15.8479
C8	125.8	84	5	12	2.5301	21.8364	2.7295	2.8604	2.7824	0.3478	0.8002	21.4335
3E-C6	118.9	72	7	14	3.0744	19.5420	2.4428	2.7494	2.7787	0.3473	0.7989	22.0645
3M-C7	118.8	76	6	14	2.8621	19.7628	2.4704	2.7607	2.7810	0.3476	0.7997	21.9365
34MM-C6	118.7	68	8	16	3.2925	18.7788	2.3474	2.7096	2.7762	0.3470	0.7979	22.3387
3E-3M-C5	118.2	64	9	18	3.5832	16.6705	2.0838	2.5905	2.7768	0.3471	0.7982	23.1188
4M-C7	117.7	75	6	14	2.9196	17.4187	2.1773	2.6344	2.7789	0.3474	0.7989	22.7102
2M-C7	117.6	79	5	14	2.7158	17.6759	2.2095	2.6491	2.7799	0.3475	0.7993	22.5967
3E-2M-C5	115.6	67	8	16	3.3549	17.4427	2.1803	2.6358	2.7789	0.3474	0.7989	22.7488
23MM-C6	115.3	70	7	16	3.1708	20.4792	2.5599	2.7963	2.7849	0.3481	0.8011	21.6556
233MMM-C5	114.6	62	9	20	3.7083	19.1115	2.3889	2.7272	2.7878	0.3485	0.8021	21.9131
234MMM-C5	113.4	65	8	18	3.4642	18.3964	2.2996	2.6890	2.7838	0.3480	0.8007	22.2412
33MM-C6	112.0	67	7	18	3.3734	18.1815	2.2727	2.6773	2.7815	0.3477	0.7999	22.3829
223MMM-C5	110.5	63	8	20	3.6233	16.8079	2.1010	2.5987	2.7851	0.3481	0.8011	22.7627
24MM-C6	109.4	71	6	16	3.0988	16.0683	2.0085	2.5537	2.7826	0.3478	0.8002	23.1970
25MM-C6	108.4	74	5	16	2.9278	18.4133	2.3017	2.6899	2.7843	0.3480	0.8009	22.2507
22MM-C6	107.0	71	5	18	3.1118	17.0338	2.1292	2.6121	2.7878	0.3485	0.8021	22.6056
2233MMMM-C4	106.0	58	9	24	4.0204	16.3152	2.0394	2.5690	2.7838	0.3480	0.8007	23.0345
224MMM-C5	99.3	66	5	20	3.3889	14.9373	1.8672	2.4807	2.7892	0.3487	0.8026	23.5670
C9	150.6	120	6	14	2.5951	27.7422	3.0825	3.2176	2.9505	0.3278	0.9766	25.9281
33EE-C5	146.2	88	12	20	3.8247	25.0208	2.7801	3.1143	2.9471	0.3275	0.9755	26.5488
3E-C7	143.0	104	8	16	3.0922	23.6799	2.6311	3.0593	2.9429	0.3270	0.9740	26.9873
3M-C8	143.0	110	7	16	2.8766	21.7527	2.4170	2.9744	2.9458	0.3273	0.9750	27.4391
4M-C8	142.5	108	7	16	2.9548	22.6204	2.5134	3.0135	2.9491	0.3277	0.9761	27.0595
2M-C8	142.5	114	6	16	2.7467	22.2705	2.4745	2.9979	2.9448	0.3272	0.9747	27.3590
3E-23MM-C5	141.6	86	12	22	3.9192	25.4119	2.8236	3.1299	2.9505	0.3278	0.9766	26.3543
2334MMMM-C5	141.5	84	12	24	4.0137	24.0988	2.6776	3.0768	2.9457	0.3273	0.9750	26.7923
4E-C7	141.2	102	8	16	3.1753	21.3349	2.3705	2.9550	2.9439	0.3271	0.9744	27.6869
3E-3M-C6	140.6	92	10	20	3.6174	22.2198	2.4689	2.9956	2.9461	0.3273	0.9751	27.2876
23MM-C7	140.5	102	8	18	3.1553	20.7438	2.3049	2.9269	2.9501	0.3278	0.9765	27.6329
334MMM-C6	140.5	88	11	22	3.8024	19.8563	2.2063	2.8832	2.9481	0.3276	0.9758	28.0907
2233MMMM-C5	140.3	82	12	26	4.1447	23.0687	2.5632	3.0331	2.9507	0.3279	0.9767	26.8886
34MM-C7	140.1	98	9	18	3.3248	22.6789	2.5199	3.0161	2.9473	0.3275	0.9755	27.1235
234MMM-C6	139.0	92	10	20	3.5758	22.6772	2.5197	3.0160	2.9482	0.3276	0.9759	27.1184
233MMM-C6	137.7	90	10	22	3.7021	20.3172	2.2575	2.9061	2.9490	0.3277	0.9761	27.8661
33MM-C7	137.3	98	8	20	3.3301	26.2722	2.9191	3.1632	2.9537	0.3282	0.9777	26.0911
3E-24MM-C5	136.7	90	10	20	3.6776	24.7896	2.7544	3.1051	2.9575	0.3286	0.9790	26.2728
35MM-C7	136.0	100	8	18	3.2230	23.9292	2.6588	3.0697	2.9542	0.3282	0.9779	26.5647
25MM-C7	136.0	104	7	18	3.0608	23.5441	2.6160	3.0535	2.9506	0.3278	0.9767	26.7803
26MM-C7	135.2	108	6	18	2.9147	21.1839	2.3538	2.9479	2.9525	0.3281	0.9773	27.4273
44MM-C7	135.2	96	8	20	3.4311	23.5541	2.6171	3.0539	2.9507	0.3279	0.9767	26.7833
4E-2M-C6	133.8	98	8	18	3.3074	22.0627	2.4514	2.9885	2.9549	0.3283	0.9781	27.0476
3E-22MM-C5	133.8	88	10	22	3.7929	21.1970	2.3552	2.9485	2.9515	0.3279	0.9770	27.4362
24MM-C7	133.5	102	7	18	3.1513	20.7945	2.3105	2.9293	2.9490	0.3277	0.9761	27.7030
2234MMMM-C5	133.0	86	10	24	3.8776	19.3005	2.1445	2.8548	2.9542	0.3283	0.9779	28.1080
22MM-C7	132.7	104	6	20	3.0730	23.9635	2.6626	3.0712	2.9540	0.3282	0.9778	26.5871
223MMM-C6	131.7	92	9	22	3.5887	22.4662	2.4962	3.0067	2.9585	0.3287	0.9793	26.8263
235MMM-C6	131.3	96	8	20	3.3766	21.6063	2.4007	2.9676	2.9548	0.3283	0.9781	27.1920
244MMM-C6	126.5	92	8	22	3.5768	20.1263	2.2363	2.8967	2.9602	0.3289	0.9799	27.5398
224MMM-C6	126.5	94	7	22	3.4673	21.2250	2.3583	2.9498	2.9512	0.3279	0.9769	27.4608
225MMM-C6	124.0	98	6	22	3.2807	19.7257	2.1917	2.8766	2.9565	0.3285	0.9786	27.8260
2244MMMM-C5	122.7	88	6	26	3.7464	18.8440	2.0938	2.8308	2.9541	0.3282	0.9779	28.3248

Reciprocal Distance – Based Indices

Another graph-invariant is the reciprocal distance matrix RD = , i,j = 1,N, where N is the total number of graph vertices. This is a symmetrical matrix whose elements are reciprocal of the topological distance [5,16,17,20,33]. The first TDIs proposed on the basis of RD have been developed by a two-steps process as follows [5,16,17].

(i)

The LOVI of each vertex in a molecular graph Γ, denoted later by μ_i, was defined from the RD using the following relation [5,16]:

(13)

In relation (13) d_ij is the topological distance between the vertices i and j, N represents the total number of vertices (i.e. non-hydrogen atoms) in Γ, and summation is made over all possible paths, from d_ij = 1 to d_ij = max(d_ij). Thus, each vertex is well characterized; it contains global information of the topological structure of Γ, the topological interaction between vertices i and j decreasing as distance d_ij is increasing. That is, for each vertex i, the quantity μ_i may be viewed as a measure if the influence of all others vertices in a given graph Γ on the vertex i.

(ii)

The LOVIs μ_i were condensed into a TDI, ^hδ, with the aid of the Randić-type formula [34], the generalized molecular connectivity [35], as follows [5,16]:

(14)

These topological distances connectivity indices (TDCIs) [5,16], also called topological distance measure connectivity indices (TDMCIs) [17], of order higher than three, have not been used in correlation due to the expected small contributions to the molecular properties.

TDCIs of order one (¹δ), two (²δ) and three (³δ) have been calculated by the following relations [5,16]:

(15)

(16)

(17)

Monoparametric correlations with molecular properties such as boiling temperatures (at normal pressure), gas chromatographic retention indices, atomization enthalpies, and molar refractions for alkanes were performed. The reported results for ²δ are very good, the correlation coefficients r being in the range 0.983 – 0.991 [5,16].

In this paper we extend TDCIs by generalization of relation (13) as follows:

(18)

Thus, we obtain a set of generalized topological distances indices (GTDIs), ^kδ_λ, where k is the same as in relation (18), which can be calculated with the following formulas:

(19)

(20)

(21)

(22)

One may easily observe that the TDMCIs in relations (15)-(17) are included in GTDIs in relations (19)-(22), and there exists a formal identity between ^λδ and ¹δ_λ (λ=1,3).

The sixteen GTDIs corresponding to in relations (19)-(22) have been calculated with the IRS computer program [31] for 72 alkanes with carbon atoms. The obtained results are given in Table 1b.

van der Waals Molecular Descriptors

No general theory of the quantitative relationship between molecular structure and molecular properties in organic chemistry (QSPR) or biological activities in medicinal chemistry (QSAR) can reasonably be regarded as satisfactory unless it provides a sound basis for predicting and interpreting linear relationships among molecular quantities.

A satisfactory theoretical model for linear correlations in organic and/or medicinal chemistry should allow reliable predictions to be made as easily as possible concerning both the circumstances in which correlations should occur (e.g., between which properties and for which compounds) and the magnitudes of the regression coefficients.

The concepts used in the model – for example, analysis into electronic (polar and resonance), hydrophobic, and steric effects – should be defined in such a way that knowledge gained through the interpretation of the linear correlations can be readily used in other areas or organic or medicinal chemistry (e.g., in elucidating the reaction mechanisms or receptor-drug molecule interaction).

Therefore, the design of molecular descriptors with very clear physical meaning is a very important task in this area of research. Analysis of the informational content of TDISs and their possible steric nature [7] as described by vdW molecular descriptors are also presented in this paper. To do this we used a set of van der Waals descriptors [3,5,6], such as the vdW molecular volume (V^W) [19,36,37,38] and surface (S^W) [18], and other descriptors of shape and size of alkane molecules, e.g. the volume of the ellipsoid which embeds the whole molecule extended along the Ox axes of Cartesian system of coordinates, V^EL, semi-axes of this ellipsoid [3,5,6], E_X, E_Y, E_Z, along Ox, Oy, and Oz axes, respectively, two measures of globularity [12], denoted by G^LOB, G^LEL and a measure of molecular packing, R^WV.

(a) Molecular van der Waals Volume

The concept of molecular volume and surface area have found many applications, not only in QSAR, but also in the studies of molecular interactions, especially in relating the bulk properties of substances like crystal packing with molecular structures [39]. The molecular volume is a measure of the space around atomic nuclei filled by electrons [40,41] and is defined geometrically as the combined volume of overlapping spheres centred on the nuclei, similar in shape to a space-filling molecular model. The van der Waals radii are used for the radii of the atomic spheres. The molecular surface area is the area of the surface that wraps the molecular volume. Exact calculation of the molecular volume and surface area is, however, a formidable task due to multiple overlap of spheres of different radii.

A molecular van der Waals envelope, ζ, can be defined in the “hard-spheres” approximation as the external surface resulted from the intersection of all vdW spheres corresponding to the atoms of molecule Μ. The points (x,y,z) inside the envelope satisfy at least one of the following inequalities:

(23)

where N represents the number of atoms of Μ. Consequently, the total volume embedded by the envelope is the molecular vdW volume ( ) of the molecule M.

Table 1b. Generalized Topological Distance Indices for the First 72 Alkanes

Alkane	1δο	2δο	3δο	4δο	1δ1	2δ1	3δ1	4δ1	1δ2	2δ2	3δ2	4δ2	1δ3	2δ3	3δ3	4δ3
C2	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	4.0000	4.0000	4.0000	4.0000
C3	5.0000	4.5000	4.2500	4.1250	2.3401	2.4960	2.5927	2.6474	2.3094	2.5298	2.6667	2.7440	5.4042	6.3143	6.9139	7.2644
C4	8.6667	7.2222	6.5741	6.2747	2.7420	3.0476	3.2273	3.3217	2.6684	3.1746	3.4867	3.6562	5.9369	7.7158	8.8912	9.5537
2-M-C3	9.0000	7.5000	6.7500	6.3750	2.6987	3.0268	3.2606	3.4058	2.4495	2.8284	3.0984	3.2660	6.6104	8.5612	10.1027	11.1232
C5	12.8333	10.0694	8.9294	8.4322	3.1512	3.6103	3.8698	4.0001	3.0184	3.8281	4.3204	4.5786	6.4421	9.1923	11.0331	12.0504
2M-C4	13.3333	10.4444	9.1481	8.5494	3.1005	3.5870	3.9085	4.0918	2.8014	3.4922	3.9664	4.2431	6.7078	9.4432	11.5347	12.8389
22MM-C3	14.0000	11.0000	9.5000	8.7500	3.0298	3.5237	3.9112	4.1707	2.5298	3.0237	3.4112	3.6707	7.6649	10.6547	13.3420	15.3090
C6	17.4000	12.9967	11.3007	10.5929	3.5580	4.1756	4.5145	4.6794	3.3552	4.4798	5.1562	5.5026	6.9256	10.6698	13.1919	14.5611
3M-C5	18.1667	13.5139	11.5775	10.7316	3.4944	4.1486	4.5609	4.7810	3.1171	4.1359	4.8312	5.2222	6.8584	10.4033	13.1037	14.7234
2M-C5	18.0000	13.4167	11.5347	10.7147	3.5062	4.1508	4.5533	4.7718	3.1536	4.1548	4.8105	5.1729	7.1146	10.8819	13.7111	15.4109
23MM-C4	18.6667	13.8889	11.7963	10.8488	3.4495	4.1170	4.5856	4.8619	2.9495	3.8299	4.4719	4.8606	7.1742	10.8019	13.8032	15.7592
22MM-C4	19.0000	14.1667	11.9722	10.9491	3.4206	4.0819	4.5653	4.8647	2.8604	3.6769	4.2923	4.6781	7.5167	11.2053	14.4129	16.6341
C7	22.3000	15.9794	13.6813	12.7551	3.9603	4.7412	5.1599	5.3589	3.6800	5.1280	5.9921	6.4268	7.3893	12.1384	15.3525	17.0740
3E-C5	23.5000	16.7083	14.0382	12.9216	3.8752	4.7084	5.2169	5.4731	3.3959	4.7553	5.6906	6.2028	6.9966	11.3769	14.7713	16.7592
3M-C6	23.2333	16.5661	13.9801	12.9001	3.8924	4.7120	5.2071	5.4618	3.4464	4.7880	5.6738	6.1525	7.2633	11.8383	15.2911	17.3058
2M-C6	22.9667	16.4239	13.9220	12.8786	3.9090	4.7153	5.1983	5.4513	3.4927	4.8103	5.6491	6.0982	7.5534	12.3402	15.8714	17.9278
23MM-C5	24.0000	17.0833	14.2569	13.0388	3.8348	4.6756	5.2385	5.5521	3.2543	4.4670	5.3366	5.8418	7.3034	11.7589	15.4142	17.7128
33MM-C5	24.5000	17.4583	14.4757	13.1560	3.7987	4.6369	5.2219	5.5613	3.1509	4.3008	5.1632	5.6840	7.4491	11.8284	15.5991	18.1020
223MMM-C4	25.0000	17.8333	14.6944	13.2731	3.7592	4.6011	5.2358	5.6326	3.0182	4.0264	4.8121	5.3131	7.7694	12.2923	16.3997	19.2855
24-MMC5	23.6667	16.8889	14.1713	13.0050	3.8562	4.6864	5.2345	5.5428	3.3081	4.4977	5.3100	5.7725	7.6854	12.5026	16.3675	18.7859
22MM-C5	24.1667	17.2639	14.3900	13.1222	3.8200	4.6450	5.2116	5.5459	3.2087	4.3428	5.1438	5.6140	7.8246	12.5789	16.5845	19.2393
C8	27.4857	19.0030	16.0677	14.9182	4.3576	5.3063	5.8056	6.0386	3.9943	5.7728	6.8275	7.3510	7.8354	13.5967	17.5119	19.5871
3E-C6	28.9667	19.8406	16.4568	15.0933	4.2637	5.2701	5.8641	6.1546	3.7022	5.3951	6.5311	7.1334	7.3777	12.7893	16.9615	19.3494
3M-C7	28.5333	19.6289	16.3767	15.0655	4.2885	5.2756	5.8524	6.1415	3.7698	5.4364	6.5110	7.0777	7.7008	13.2928	17.4546	19.8252
34MM-C6	29.7333	20.3578	16.7336	15.2320	4.2109	5.2319	5.8921	6.2430	3.5361	5.0896	6.1974	6.8225	7.4466	12.7169	17.0353	19.6753
3E-3M-C5	30.5000	20.8750	17.0104	15.3707	4.1627	5.1871	5.8802	6.2603	3.4063	4.8953	6.0230	6.6881	7.4129	12.4814	16.8736	19.6994
4M-C7	28.6333	19.6739	16.3920	15.0702	4.2834	5.2746	5.8538	6.1428	3.7575	5.4330	6.5155	7.0829	7.6393	13.2499	17.4739	19.8881
2M-C7	28.2000	19.4622	16.3119	15.0424	4.3072	5.2797	5.8435	6.1309	3.8189	5.4601	6.4857	7.0227	7.9891	13.7942	18.0287	20.4407
3E-2M-C5	29.8333	20.4028	16.7488	15.2366	4.2051	5.2305	5.8943	6.2452	3.5201	5.0765	6.1945	6.8241	7.3998	12.6984	17.1022	19.8072
23MM-C6	29.4667	20.2156	16.6755	15.2105	4.2265	5.2371	5.8846	6.2330	3.5770	5.1171	6.1800	6.7729	7.6750	13.1693	17.5974	20.2978
233MMM-C5	31.0000	21.2500	17.2292	15.4878	4.1270	5.1511	5.8919	6.3299	3.2923	4.6369	5.6781	6.3186	7.7163	12.9264	17.6277	20.8159
234MMM-C5	30.3333	20.7778	16.9676	15.3538	4.1686	5.1968	5.9131	6.3223	3.3990	4.8047	5.8459	6.4638	7.6951	13.0744	17.7163	20.7199
33MM-C6	30.0667	20.6356	16.9095	15.3323	4.1883	5.1979	5.8690	6.2431	3.4731	4.9521	6.0113	6.6197	7.7662	13.1951	17.7780	20.7155
223MMM-C5	30.8333	21.1528	17.1863	15.4710	4.1365	5.1563	5.8887	6.3236	3.3155	4.6580	5.6766	6.2961	7.8598	13.2227	18.0216	21.2749
24MM-C6	29.3000	20.1183	16.6327	15.1936	4.2360	5.2451	5.8878	6.2329	3.5965	5.1282	6.1730	6.7523	7.8211	13.4491	17.9496	20.6859
25MM-C6	28.9333	19.9311	16.5594	15.1675	4.2564	5.2519	5.8805	6.2228	3.6463	5.1511	6.1455	6.6950	8.1313	13.9817	18.5400	21.2935
22MM-C6	29.5333	20.3511	16.7934	15.2893	4.2183	5.2087	5.8567	6.2256	3.5484	5.0012	5.9848	6.5405	8.2150	14.0130	18.7422	21.7600
2233MMMM-C4	32.0000	22.0000	17.6667	15.7222	4.0599	5.0746	5.8780	6.3994	3.0987	4.2357	5.1633	5.7769	8.1946	13.5658	18.7685	22.6023
224MMM-C5	30.3333	20.8611	17.0579	15.4203	4.1649	5.1758	5.8899	6.3159	3.3789	4.6995	5.6511	6.2177	8.3047	14.1290	19.2053	22.6104
C9	32.9214	22.0579	18.4580	17.0818	4.7502	5.8708	6.4512	6.7183	4.2996	6.4144	7.6624	8.2752	8.2660	15.0456	19.6696	22.1000
33EE-C5	37.0000	24.4167	19.5764	17.5932	4.5127	5.7311	6.5397	6.9614	3.6315	5.4611	6.8712	7.6900	7.3862	13.1373	18.2140	21.4143
3E-C7	34.6000	22.9589	18.8626	17.2603	4.6523	5.8320	6.5097	6.8346	4.0103	6.0358	7.3669	8.0585	7.8024	14.2310	19.1250	21.8706
3M-C8	34.0524	22.7080	18.7724	17.2302	4.6811	5.8389	6.4977	6.8212	4.0843	6.0809	7.3466	8.0021	8.1361	14.7428	19.6123	22.3387
4M-C8	34.2190	22.7775	18.7944	17.2364	4.6733	5.8372	6.4993	6.8226	4.0669	6.0763	7.3521	8.0080	8.0534	14.6879	19.6343	22.4074
2M-C8	33.6714	22.5266	18.7041	17.2063	4.7009	5.8437	6.4889	6.8105	4.1340	6.1054	7.3213	7.9470	8.4147	15.2408	20.1847	22.9532
3E-23MM-C5	37.5000	24.7917	19.7951	17.7104	4.4802	5.6951	6.5492	7.0295	3.5326	5.2173	6.5325	7.3221	7.6733	13.5674	18.9279	22.4681
2334MMMM-C5	38.0000	25.1667	20.0139	17.8275	4.4477	5.6592	6.5587	7.0975	3.4337	4.9735	6.1938	6.9541	7.9605	14.0001	19.6539	23.5480
4E-C7	34.7667	23.0283	18.8846	17.2666	4.6441	5.8303	6.5118	6.8364	3.9904	6.0269	7.3706	8.0640	7.7226	14.1724	19.1455	21.9393
3E-3M-C6	36.4667	24.1322	19.4602	17.5502	4.5420	5.7452	6.5280	6.9427	3.7049	5.5319	6.8675	7.6236	7.7215	13.8285	19.0539	22.3194
23MM-C7	35.1000	23.3339	19.0814	17.3775	4.6176	5.7992	6.5297	6.9128	3.8964	5.7645	7.0174	7.6982	8.0909	14.6099	19.7581	22.8174
334MMM-C6	37.2333	24.6494	19.7371	17.6889	4.4945	5.7029	6.5450	7.0214	3.5669	5.2523	6.5375	7.3006	7.8283	13.8587	19.2586	22.8134
2233MMMM-C5	38.5000	25.5417	20.2326	17.9447	4.4191	5.6198	6.5332	7.0972	3.3628	4.8363	6.0255	6.7821	8.1344	14.1916	20.0131	24.1679
34MM-C7	35.5333	23.5456	19.1614	17.4052	4.5950	5.7918	6.5385	6.9241	3.8412	5.7309	7.0389	7.7534	7.8077	14.1126	19.2166	22.2610
234MMM-C6	36.4667	24.1322	19.4602	17.5502	4.5377	5.7501	6.5661	7.0132	3.6731	5.4219	6.7052	7.4444	7.8294	14.0204	19.3366	22.6855
233MMM-C6	36.9667	24.5072	19.6790	17.6674	4.5090	5.7095	6.5388	7.0119	3.6029	5.2825	6.5257	7.2548	8.0118	14.2671	19.7998	23.4304
33MM-C7	35.7667	23.7783	19.3221	17.5009	4.5789	5.7599	6.5143	6.9230	3.7952	5.6017	6.8504	7.5460	8.1623	14.6262	19.9391	23.2389
3E-24MM-C5	36.6667	24.2222	19.4907	17.5594	4.5272	5.7459	6.5684	7.0162	3.6479	5.4013	6.7009	7.4473	7.7455	13.9657	19.4139	22.8677
35MM-C7	35.2667	23.4033	19.1034	17.3837	4.6087	5.8019	6.5413	6.9232	3.8696	5.7494	7.0335	7.7318	7.9742	14.4001	19.5371	22.5891
25MM-C7	34.8333	23.1917	19.0233	17.3560	4.6310	5.8100	6.5342	6.9130	3.9217	5.7757	7.0071	7.6745	8.2729	14.9280	20.1221	23.1915
26MM-C7	34.4333	23.0006	18.9517	17.3312	4.6515	5.8159	6.5261	6.9026	3.9707	5.7989	6.9809	7.6190	8.5582	15.4324	20.6972	23.8055
44MM-C7	35.9667	23.8683	19.3526	17.5102	4.5695	5.7574	6.5165	6.9252	3.7743	5.5942	6.8580	7.5555	8.0562	14.5324	19.9493	23.3281
4E-2M-C6	35.4333	23.4728	19.1253	17.3900	4.5999	5.7998	6.5442	6.9259	3.8467	5.7313	7.0295	7.7335	7.9109	14.3765	19.6153	22.7325
3E-22MM-C5	37.1667	24.5972	19.7095	17.6766	4.4979	5.7064	6.5440	7.0172	3.5737	5.2601	6.5330	7.2796	7.9123	14.1177	19.7050	23.3971
24MM-C7	35.0333	23.2817	19.0538	17.3652	4.6216	5.8060	6.5343	6.9139	3.9032	5.7719	7.0149	7.6829	8.1669	14.8378	20.1251	23.2669
2234MMMM-C5	37.6667	24.9722	19.9282	17.7938	4.4646	5.6723	6.5601	7.0926	3.4691	5.0036	6.1899	6.9201	8.1948	14.4912	20.3010	24.2834
22MM-C7	35.1000	23.4450	19.1925	17.4546	4.6130	5.7721	6.5017	6.9051	3.8764	5.6527	6.8222	7.4653	8.6219	15.4526	20.8955	24.2724
223MMM-C6	36.7000	24.3650	19.6209	17.6459	4.5226	5.7161	6.5348	7.0046	3.6339	5.3072	6.5210	7.2279	8.1914	14.6038	20.1965	23.8601
235MMM-C6	35.9333	23.8478	19.3441	17.5072	4.5656	5.7667	6.5637	7.0037	3.7352	5.4631	6.6812	7.3734	8.1979	14.7547	20.2447	23.6759
244MMM-C6	36.6333	24.3128	19.5933	17.6336	4.5258	5.7253	6.5466	7.0132	3.6349	5.3030	6.5169	7.2233	8.2476	14.7385	20.4006	24.0912
224MMM-C6	36.3667	24.1706	19.5353	17.6121	4.5394	5.7322	6.5428	7.0060	3.6656	5.3281	6.5138	7.1978	8.4249	15.0656	20.7876	24.5142
225MMM-C6	35.9000	23.9383	19.4467	17.5814	4.5628	5.7423	6.5373	6.9966	3.7170	5.3520	6.4847	7.1383	8.7454	15.6258	21.3973	25.1226
2244MMMM-C5	37.5000	24.9583	19.9757	17.8435	4.4696	5.6607	6.5422	7.0877	3.4658	4.9149	5.9990	6.6664	8.8457	15.6909	22.0017	26.4205

Table 1b. Generalized Topological Distance Indices for the First 72 Alkanes

Alkane	1δο	2δο	3δο	4δο	1δ1	2δ1	3δ1	4δ1	1δ2	2δ2	3δ2	4δ2	1δ3	2δ3	3δ3	4δ3
C2	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	2.0000	4.0000	4.0000	4.0000	4.0000
C3	5.0000	4.5000	4.2500	4.1250	2.3401	2.4960	2.5927	2.6474	2.3094	2.5298	2.6667	2.7440	5.4042	6.3143	6.9139	7.2644
C4	8.6667	7.2222	6.5741	6.2747	2.7420	3.0476	3.2273	3.3217	2.6684	3.1746	3.4867	3.6562	5.9369	7.7158	8.8912	9.5537
2-M-C3	9.0000	7.5000	6.7500	6.3750	2.6987	3.0268	3.2606	3.4058	2.4495	2.8284	3.0984	3.2660	6.6104	8.5612	10.1027	11.1232
C5	12.8333	10.0694	8.9294	8.4322	3.1512	3.6103	3.8698	4.0001	3.0184	3.8281	4.3204	4.5786	6.4421	9.1923	11.0331	12.0504
2M-C4	13.3333	10.4444	9.1481	8.5494	3.1005	3.5870	3.9085	4.0918	2.8014	3.4922	3.9664	4.2431	6.7078	9.4432	11.5347	12.8389
22MM-C3	14.0000	11.0000	9.5000	8.7500	3.0298	3.5237	3.9112	4.1707	2.5298	3.0237	3.4112	3.6707	7.6649	10.6547	13.3420	15.3090
C6	17.4000	12.9967	11.3007	10.5929	3.5580	4.1756	4.5145	4.6794	3.3552	4.4798	5.1562	5.5026	6.9256	10.6698	13.1919	14.5611
3M-C5	18.1667	13.5139	11.5775	10.7316	3.4944	4.1486	4.5609	4.7810	3.1171	4.1359	4.8312	5.2222	6.8584	10.4033	13.1037	14.7234
2M-C5	18.0000	13.4167	11.5347	10.7147	3.5062	4.1508	4.5533	4.7718	3.1536	4.1548	4.8105	5.1729	7.1146	10.8819	13.7111	15.4109
23MM-C4	18.6667	13.8889	11.7963	10.8488	3.4495	4.1170	4.5856	4.8619	2.9495	3.8299	4.4719	4.8606	7.1742	10.8019	13.8032	15.7592
22MM-C4	19.0000	14.1667	11.9722	10.9491	3.4206	4.0819	4.5653	4.8647	2.8604	3.6769	4.2923	4.6781	7.5167	11.2053	14.4129	16.6341
C7	22.3000	15.9794	13.6813	12.7551	3.9603	4.7412	5.1599	5.3589	3.6800	5.1280	5.9921	6.4268	7.3893	12.1384	15.3525	17.0740
3E-C5	23.5000	16.7083	14.0382	12.9216	3.8752	4.7084	5.2169	5.4731	3.3959	4.7553	5.6906	6.2028	6.9966	11.3769	14.7713	16.7592
3M-C6	23.2333	16.5661	13.9801	12.9001	3.8924	4.7120	5.2071	5.4618	3.4464	4.7880	5.6738	6.1525	7.2633	11.8383	15.2911	17.3058
2M-C6	22.9667	16.4239	13.9220	12.8786	3.9090	4.7153	5.1983	5.4513	3.4927	4.8103	5.6491	6.0982	7.5534	12.3402	15.8714	17.9278
23MM-C5	24.0000	17.0833	14.2569	13.0388	3.8348	4.6756	5.2385	5.5521	3.2543	4.4670	5.3366	5.8418	7.3034	11.7589	15.4142	17.7128
33MM-C5	24.5000	17.4583	14.4757	13.1560	3.7987	4.6369	5.2219	5.5613	3.1509	4.3008	5.1632	5.6840	7.4491	11.8284	15.5991	18.1020
223MMM-C4	25.0000	17.8333	14.6944	13.2731	3.7592	4.6011	5.2358	5.6326	3.0182	4.0264	4.8121	5.3131	7.7694	12.2923	16.3997	19.2855
24-MMC5	23.6667	16.8889	14.1713	13.0050	3.8562	4.6864	5.2345	5.5428	3.3081	4.4977	5.3100	5.7725	7.6854	12.5026	16.3675	18.7859
22MM-C5	24.1667	17.2639	14.3900	13.1222	3.8200	4.6450	5.2116	5.5459	3.2087	4.3428	5.1438	5.6140	7.8246	12.5789	16.5845	19.2393
C8	27.4857	19.0030	16.0677	14.9182	4.3576	5.3063	5.8056	6.0386	3.9943	5.7728	6.8275	7.3510	7.8354	13.5967	17.5119	19.5871
3E-C6	28.9667	19.8406	16.4568	15.0933	4.2637	5.2701	5.8641	6.1546	3.7022	5.3951	6.5311	7.1334	7.3777	12.7893	16.9615	19.3494
3M-C7	28.5333	19.6289	16.3767	15.0655	4.2885	5.2756	5.8524	6.1415	3.7698	5.4364	6.5110	7.0777	7.7008	13.2928	17.4546	19.8252
34MM-C6	29.7333	20.3578	16.7336	15.2320	4.2109	5.2319	5.8921	6.2430	3.5361	5.0896	6.1974	6.8225	7.4466	12.7169	17.0353	19.6753
3E-3M-C5	30.5000	20.8750	17.0104	15.3707	4.1627	5.1871	5.8802	6.2603	3.4063	4.8953	6.0230	6.6881	7.4129	12.4814	16.8736	19.6994
4M-C7	28.6333	19.6739	16.3920	15.0702	4.2834	5.2746	5.8538	6.1428	3.7575	5.4330	6.5155	7.0829	7.6393	13.2499	17.4739	19.8881
2M-C7	28.2000	19.4622	16.3119	15.0424	4.3072	5.2797	5.8435	6.1309	3.8189	5.4601	6.4857	7.0227	7.9891	13.7942	18.0287	20.4407
3E-2M-C5	29.8333	20.4028	16.7488	15.2366	4.2051	5.2305	5.8943	6.2452	3.5201	5.0765	6.1945	6.8241	7.3998	12.6984	17.1022	19.8072
23MM-C6	29.4667	20.2156	16.6755	15.2105	4.2265	5.2371	5.8846	6.2330	3.5770	5.1171	6.1800	6.7729	7.6750	13.1693	17.5974	20.2978
233MMM-C5	31.0000	21.2500	17.2292	15.4878	4.1270	5.1511	5.8919	6.3299	3.2923	4.6369	5.6781	6.3186	7.7163	12.9264	17.6277	20.8159
234MMM-C5	30.3333	20.7778	16.9676	15.3538	4.1686	5.1968	5.9131	6.3223	3.3990	4.8047	5.8459	6.4638	7.6951	13.0744	17.7163	20.7199
33MM-C6	30.0667	20.6356	16.9095	15.3323	4.1883	5.1979	5.8690	6.2431	3.4731	4.9521	6.0113	6.6197	7.7662	13.1951	17.7780	20.7155
223MMM-C5	30.8333	21.1528	17.1863	15.4710	4.1365	5.1563	5.8887	6.3236	3.3155	4.6580	5.6766	6.2961	7.8598	13.2227	18.0216	21.2749
24MM-C6	29.3000	20.1183	16.6327	15.1936	4.2360	5.2451	5.8878	6.2329	3.5965	5.1282	6.1730	6.7523	7.8211	13.4491	17.9496	20.6859
25MM-C6	28.9333	19.9311	16.5594	15.1675	4.2564	5.2519	5.8805	6.2228	3.6463	5.1511	6.1455	6.6950	8.1313	13.9817	18.5400	21.2935
22MM-C6	29.5333	20.3511	16.7934	15.2893	4.2183	5.2087	5.8567	6.2256	3.5484	5.0012	5.9848	6.5405	8.2150	14.0130	18.7422	21.7600
2233MMMM-C4	32.0000	22.0000	17.6667	15.7222	4.0599	5.0746	5.8780	6.3994	3.0987	4.2357	5.1633	5.7769	8.1946	13.5658	18.7685	22.6023
224MMM-C5	30.3333	20.8611	17.0579	15.4203	4.1649	5.1758	5.8899	6.3159	3.3789	4.6995	5.6511	6.2177	8.3047	14.1290	19.2053	22.6104
C9	32.9214	22.0579	18.4580	17.0818	4.7502	5.8708	6.4512	6.7183	4.2996	6.4144	7.6624	8.2752	8.2660	15.0456	19.6696	22.1000
33EE-C5	37.0000	24.4167	19.5764	17.5932	4.5127	5.7311	6.5397	6.9614	3.6315	5.4611	6.8712	7.6900	7.3862	13.1373	18.2140	21.4143
3E-C7	34.6000	22.9589	18.8626	17.2603	4.6523	5.8320	6.5097	6.8346	4.0103	6.0358	7.3669	8.0585	7.8024	14.2310	19.1250	21.8706
3M-C8	34.0524	22.7080	18.7724	17.2302	4.6811	5.8389	6.4977	6.8212	4.0843	6.0809	7.3466	8.0021	8.1361	14.7428	19.6123	22.3387
4M-C8	34.2190	22.7775	18.7944	17.2364	4.6733	5.8372	6.4993	6.8226	4.0669	6.0763	7.3521	8.0080	8.0534	14.6879	19.6343	22.4074
2M-C8	33.6714	22.5266	18.7041	17.2063	4.7009	5.8437	6.4889	6.8105	4.1340	6.1054	7.3213	7.9470	8.4147	15.2408	20.1847	22.9532
3E-23MM-C5	37.5000	24.7917	19.7951	17.7104	4.4802	5.6951	6.5492	7.0295	3.5326	5.2173	6.5325	7.3221	7.6733	13.5674	18.9279	22.4681
2334MMMM-C5	38.0000	25.1667	20.0139	17.8275	4.4477	5.6592	6.5587	7.0975	3.4337	4.9735	6.1938	6.9541	7.9605	14.0001	19.6539	23.5480
4E-C7	34.7667	23.0283	18.8846	17.2666	4.6441	5.8303	6.5118	6.8364	3.9904	6.0269	7.3706	8.0640	7.7226	14.1724	19.1455	21.9393
3E-3M-C6	36.4667	24.1322	19.4602	17.5502	4.5420	5.7452	6.5280	6.9427	3.7049	5.5319	6.8675	7.6236	7.7215	13.8285	19.0539	22.3194
23MM-C7	35.1000	23.3339	19.0814	17.3775	4.6176	5.7992	6.5297	6.9128	3.8964	5.7645	7.0174	7.6982	8.0909	14.6099	19.7581	22.8174
334MMM-C6	37.2333	24.6494	19.7371	17.6889	4.4945	5.7029	6.5450	7.0214	3.5669	5.2523	6.5375	7.3006	7.8283	13.8587	19.2586	22.8134
2233MMMM-C5	38.5000	25.5417	20.2326	17.9447	4.4191	5.6198	6.5332	7.0972	3.3628	4.8363	6.0255	6.7821	8.1344	14.1916	20.0131	24.1679
34MM-C7	35.5333	23.5456	19.1614	17.4052	4.5950	5.7918	6.5385	6.9241	3.8412	5.7309	7.0389	7.7534	7.8077	14.1126	19.2166	22.2610
234MMM-C6	36.4667	24.1322	19.4602	17.5502	4.5377	5.7501	6.5661	7.0132	3.6731	5.4219	6.7052	7.4444	7.8294	14.0204	19.3366	22.6855
233MMM-C6	36.9667	24.5072	19.6790	17.6674	4.5090	5.7095	6.5388	7.0119	3.6029	5.2825	6.5257	7.2548	8.0118	14.2671	19.7998	23.4304
33MM-C7	35.7667	23.7783	19.3221	17.5009	4.5789	5.7599	6.5143	6.9230	3.7952	5.6017	6.8504	7.5460	8.1623	14.6262	19.9391	23.2389
3E-24MM-C5	36.6667	24.2222	19.4907	17.5594	4.5272	5.7459	6.5684	7.0162	3.6479	5.4013	6.7009	7.4473	7.7455	13.9657	19.4139	22.8677
35MM-C7	35.2667	23.4033	19.1034	17.3837	4.6087	5.8019	6.5413	6.9232	3.8696	5.7494	7.0335	7.7318	7.9742	14.4001	19.5371	22.5891
25MM-C7	34.8333	23.1917	19.0233	17.3560	4.6310	5.8100	6.5342	6.9130	3.9217	5.7757	7.0071	7.6745	8.2729	14.9280	20.1221	23.1915
26MM-C7	34.4333	23.0006	18.9517	17.3312	4.6515	5.8159	6.5261	6.9026	3.9707	5.7989	6.9809	7.6190	8.5582	15.4324	20.6972	23.8055
44MM-C7	35.9667	23.8683	19.3526	17.5102	4.5695	5.7574	6.5165	6.9252	3.7743	5.5942	6.8580	7.5555	8.0562	14.5324	19.9493	23.3281
4E-2M-C6	35.4333	23.4728	19.1253	17.3900	4.5999	5.7998	6.5442	6.9259	3.8467	5.7313	7.0295	7.7335	7.9109	14.3765	19.6153	22.7325
3E-22MM-C5	37.1667	24.5972	19.7095	17.6766	4.4979	5.7064	6.5440	7.0172	3.5737	5.2601	6.5330	7.2796	7.9123	14.1177	19.7050	23.3971
24MM-C7	35.0333	23.2817	19.0538	17.3652	4.6216	5.8060	6.5343	6.9139	3.9032	5.7719	7.0149	7.6829	8.1669	14.8378	20.1251	23.2669
2234MMMM-C5	37.6667	24.9722	19.9282	17.7938	4.4646	5.6723	6.5601	7.0926	3.4691	5.0036	6.1899	6.9201	8.1948	14.4912	20.3010	24.2834
22MM-C7	35.1000	23.4450	19.1925	17.4546	4.6130	5.7721	6.5017	6.9051	3.8764	5.6527	6.8222	7.4653	8.6219	15.4526	20.8955	24.2724
223MMM-C6	36.7000	24.3650	19.6209	17.6459	4.5226	5.7161	6.5348	7.0046	3.6339	5.3072	6.5210	7.2279	8.1914	14.6038	20.1965	23.8601
235MMM-C6	35.9333	23.8478	19.3441	17.5072	4.5656	5.7667	6.5637	7.0037	3.7352	5.4631	6.6812	7.3734	8.1979	14.7547	20.2447	23.6759
244MMM-C6	36.6333	24.3128	19.5933	17.6336	4.5258	5.7253	6.5466	7.0132	3.6349	5.3030	6.5169	7.2233	8.2476	14.7385	20.4006	24.0912
224MMM-C6	36.3667	24.1706	19.5353	17.6121	4.5394	5.7322	6.5428	7.0060	3.6656	5.3281	6.5138	7.1978	8.4249	15.0656	20.7876	24.5142
225MMM-C6	35.9000	23.9383	19.4467	17.5814	4.5628	5.7423	6.5373	6.9966	3.7170	5.3520	6.4847	7.1383	8.7454	15.6258	21.3973	25.1226
2244MMMM-C5	37.5000	24.9583	19.9757	17.8435	4.4696	5.6607	6.5422	7.0877	3.4658	4.9149	5.9990	6.6664	8.8457	15.6909	22.0017	26.4205

The following integral:

(24)

can be intuitively justified as a volume [3,5,19]. This assumption is natural because the properties of molecular vdW space can be considered independent from the nature of the atoms, even in the case when domains of the vdW atomic spheres intersect.

To estimate the integral (24), the molecule (23) is inserted into a bounding parallelepiped with the volume V_p. The random points are generated into the parallelepiped, which includes the domain M. If n_t is the total number of generated points and n_s the number of points that which satisfy the inequalities in (23) than the van der Waals volume is:

(25)

In order to avoid multiple computation of the same volume, resulted from multiple atom spheres overlapping, we applied a Monte Carlo technique [42]. The accuracy ε of the estimate (25) for a given maximum probability, δ, is inversely proportional to the square root of the number of trials, or

(26)

Taking into consideration the precision and the accuracy of chemical and biological experiments, for ε = 0.05 and δ = 0.01, the number of necessary points is N = 10,000. This makes the Monte Carlo method not difficult to apply, due to the performances of nowadays computers. In order to increase the accuracy of the method the calculus must be repeated at least 10 to 20 times for each calculated volume. The final result, i.e. the mean value of these computed volumes, is validated by statistical methods.

(b) Molecular van der Waals Surface

The van der Waals volume of the envelope, ζ, defined in the previous paragraph, can be a measure of the molecules’ size. Obviously, this envelope is a surface, and there were methods developed to compute the area of this surface [5,18,42,43]. Some of them are based on a Monte Carlo method [5,18], others on an analytical algorithm [44]. The computed surfaces were especially used to characterize the shape and the similarity of the molecules, their graphical representation [44,45], and so on. A Monte Carlo algorithm [5,18] implies the generation of an uniform grid on each sphere of the molecule, followed by the detection of the number of points generated on the surface (n_t) and of those (n_e) that do not satisfy the inequalities in (23). For every “hard sphere” i, one computes the outer part of each sphere’s surface, :

(27)

The final surface is computed as a sum of exterior surface of each sphere, :

(28)

See refs. [5,18] for details about how to generate a uniform grid.

(c) Synthetic van der Waals descriptors of molecular shape

The shape of molecules is doubtlessly the main element of most chemical interactions. Quantitative treatment of molecular shape, that is the development of appropriate molecular descriptors able to synthesize the characteristics of 3D extension of molecules, is a very difficult problem. Most procedures are based either on comparing molecules with a reference structure, or on dividing them and defining the sectors by means of Euclidean distance between certain atoms or with the aid of Cartesian coordinates of those sectors.

Using a hard-spheres model, we developed a series of van der Waals indicators of the molecular shape. This model allowed the introduction of several synthetic descriptors of molecular shape, which are presented as follows.

A first set of indicators was developed starting from the fact that a molecule can be characterized by the surface of molecular envelope described by equations (23) (with the sign “=”).

The equation (29) represents a 2^nd -degree equation describing a general surface [5]:

a₁₁∙x² + a₂₂∙y² + a₃₃∙z² + 2a₁₄ + 2a₂₄∙y 2a₃₄∙z + a₄₄ = 0

(29)

By transformations of coordinates (translation), equation (29) is simplified and reduced to one of the fifteen equations composed of four terms [46]. For obvious physical reasons related to spatial extension of substituents, we neglected both singular quadrics and the equations that do not have real solutions – and, therefore, do not represent geometrical figures. From the five non-singular surfaces of 2^nd degree which remain and represent geometrical figures (ellipsoid, ellipsoidal and hyperbolic paraboloid, and one-sheet and two-sheet hyperboloid), only the ellipsoid fulfils the physical conditions so that by assimilating the molecule with this geometrical figure the physical meaning of the calculated parameters is maintained.

It is known that the relationship:

(30)

represents an ellipsoid, namely a spheroid (or conoid). If E_X < E_Y = E_Z equation (30) represents a prolate ellipsoid. If E_X = E_Y > E_Z the relationship (30) represent an oblate ellipsoid of revolution, and if E_X = E_Y = E_Z we have a sphere.

The molecules are oriented along the Ox axes of the Cartesian coordinate system and the volume of the ellipsoid (30) and its vdW centre are estimated by a Monte Carlo algorithm implemented in the IRS computer program [31]. Then, the semi-axes of the ellipsoid are calculated.

Starting from the concept of packing density and from the fact that the experimental determination of the cross-section area of a molecule [47] is performed by assimilating it to a sphere, and assuming a maximal packing of molecule spheres, one may consider as a quantitative measure of the steric characteristics of molecules the descriptor R^WV, defined as follows:

(31)

where V^W and S^W are the calculated vdW volume and surface, respectively (see above the corresponding sections)

(d) Globularity measures

With the help of the molecular vdW descriptors computed, two other parameters can be defined. The first is defined only for acyclic molecules, named globularity measure (G^LOB), and is given by relation [5,12]:

(32)

where R^WV is defined by relation (31) and R_s represents the ratio between the volume and the surface of an equivalent sphere, which surrounds the molecule, with the radius equal to the half of the longest dimension of the parallelepiped that embeds the molecule. The above relation cannot be used for cyclic molecules, because the volume of the equivalent sphere includes the internal empty space, which is not included in the van der Waals volume.

The second one is defined by the following equation:

(33)

where V^EL is the volume of the ellipsoid surrounding the whole molecule, and V^S is the volume of a sphere with a radius equal to half of the longest ellipsoid axe. This parameter should be more useful for characterizing globularity because it includes the volume of all holes which may appear.

These two parameters can be used to describe the shape of acyclic molecules. The globularity measure decreases with the growth of the linear chains and increases toward unity when the molecule is highly branched or compacted.

Table 2. Van der Waals Molecular Descriptors of the First 72 Alkanes

Alkane	VW	SW	VEL	EX	EY	EZ	GLOB	GLEL	RWV
C2	45.672	71.059	114.658	3.021	3.145	2.881	0.613	0.880	0.643
C3	62.678	93.293	160.367	3.516	3.780	2.881	0.533	0.709	0.672
C4	79.733	115.287	190.706	3.763	4.200	2.881	0.494	0.615	0.692
2-M-C3	79.699	114.746	217.825	3.889	3.781	3.536	0.536	0.884	0.695
C5	96.825	137.555	247.324	4.252	4.820	2.881	0.438	0.527	0.704
2M-C4	96.719	135.905	264.984	4.182	4.248	3.561	0.503	0.825	0.712
22MM-C3	96.112	135.069	254.783	3.875	3.766	4.168	0.512	0.840	0.712
C6	113.685	159.426	284.774	4.503	5.241	2.881	0.408	0.472	0.713
3M-C5	113.597	156.104	318.161	4.307	4.899	3.599	0.446	0.646	0.728
2M-C5	113.775	157.987	339.431	4.687	4.811	3.594	0.449	0.728	0.720
23MM-C4	113.295	154.842	332.325	4.503	4.220	4.175	0.488	0.869	0.732
22MM-C4	113.230	155.351	311.724	4.215	4.235	4.168	0.516	0.979	0.729
C7	131.027	181.788	352.736	4.988	5.861	2.881	0.369	0.418	0.721
3E-C5	130.432	176.805	390.419	4.661	4.931	4.055	0.449	0.777	0.738
3M-C6	130.657	178.137	369.805	4.503	5.352	3.664	0.411	0.576	0.733
2M-C6	130.675	180.112	389.158	4.940	5.265	3.572	0.413	0.637	0.726
23MM-C5	130.175	174.531	397.190	4.681	4.859	4.168	0.460	0.826	0.746
33MM-C5	130.438	173.979	373.215	4.344	4.921	4.168	0.457	0.748	0.750
223MMM-C4	130.086	173.065	340.481	4.587	4.251	4.169	0.492	0.842	0.752
24-MMC5	130.472	177.729	363.282	4.746	5.027	3.635	0.438	0.683	0.734
22MM-C5	130.431	177.235	390.649	4.663	4.799	4.168	0.460	0.844	0.736
C8	147.814	204.050	397.254	5.240	6.281	2.881	0.346	0.383	0.724
3E-C6	147.572	199.053	423.503	4.662	5.378	4.033	0.414	0.650	0.741
3M-C7	147.843	200.529	459.253	5.026	5.961	3.660	0.371	0.518	0.737
34MM-C6	146.984	194.367	422.458	4.536	5.351	4.156	0.424	0.658	0.756
3E-3M-C5	147.192	193.627	428.593	4.792	4.950	4.314	0.461	0.844	0.760
4M-C7	147.596	200.355	450.614	4.972	5.854	3.696	0.378	0.536	0.737
2M-C7	147.868	201.974	478.805	5.432	5.861	3.591	0.375	0.568	0.732
3E-2M-C5	147.146	193.534	427.292	4.946	4.910	4.201	0.461	0.843	0.760
23MM-C6	147.319	196.670	452.909	4.852	5.351	4.165	0.420	0.706	0.749
233MMM-C5	147.226	191.854	411.251	4.769	4.944	4.164	0.466	0.813	0.767
234MMM-C5	147.420	194.534	397.214	4.780	4.962	3.998	0.458	0.776	0.758
33MM-C6	147.539	195.769	422.747	4.523	5.353	4.168	0.422	0.658	0.754
223MMM-C5	147.123	193.437	401.790	4.740	4.854	4.169	0.470	0.839	0.761
25MM-C6	147.027	200.288	492.840	5.286	5.260	4.232	0.417	0.797	0.734
22MM-C6	147.747	199.191	456.496	4.961	5.270	4.168	0.422	0.744	0.742
2233MMMM-C4	146.856	188.312	338.229	4.563	4.239	4.175	0.513	0.850	0.780
224MMM-C5	147.018	196.564	410.699	4.714	5.036	4.130	0.446	0.768	0.748
C9	164.641	226.192	476.615	5.723	6.901	2.881	0.316	0.346	0.728
33EE-C5	164.061	211.324	459.204	4.769	5.035	4.566	0.463	0.859	0.776
3E-C7	164.668	221.159	532.315	5.028	6.010	4.206	0.372	0.585	0.745
3M-C8	164.821	222.363	518.339	5.259	6.396	3.680	0.348	0.473	0.741
4M-C8	164.737	222.348	518.784	5.214	6.354	3.739	0.350	0.483	0.741
2M-C8	164.651	224.148	535.314	5.679	6.300	3.572	0.350	0.511	0.735
3E-23MM-C5	163.900	210.954	455.680	5.057	5.038	4.269	0.461	0.841	0.777
2334MMMM-C5	163.486	209.776	419.044	4.812	5.003	4.155	0.467	0.799	0.779
4E-C7	164.814	220.815	487.239	5.054	5.713	4.029	0.392	0.624	0.746
3E-3M-C6	164.119	215.689	472.306	4.804	5.410	4.338	0.422	0.712	0.761
23MM-C7	164.329	218.824	558.982	5.408	5.934	4.158	0.380	0.639	0.751
334MMM-C6	163.840	212.499	440.007	4.587	5.505	4.160	0.420	0.629	0.771
2233MMMM-C5	163.704	205.939	400.956	4.676	4.902	4.176	0.486	0.813	0.795
34MM-C7	163.970	215.894	522.987	5.081	5.921	4.150	0.385	0.601	0.759
234MMM-C6	164.105	214.473	460.278	4.964	5.428	4.078	0.423	0.687	0.765
233MMM-C6	164.180	213.892	455.137	4.831	5.398	4.167	0.427	0.691	0.768
33MM-C7	164.007	217.971	529.200	5.085	5.962	4.168	0.379	0.596	0.752
3E-24MM-C5	164.363	216.103	427.188	5.057	5.064	3.982	0.451	0.785	0.761
35MM-C7	164.491	218.873	533.826	5.433	5.307	4.420	0.415	0.795	0.752
25MM-C7	164.140	220.135	568.077	5.393	5.886	4.272	0.380	0.665	0.746
26MM-C7	164.715	222.197	491.514	5.443	5.995	3.596	0.371	0.545	0.741
44MM-C7	164.587	217.682	501.828	4.898	5.869	4.168	0.386	0.593	0.756
4E-2M-C6	164.549	219.575	436.658	5.125	5.414	3.757	0.415	0.657	0.749
3E-22MM-C5	164.151	214.491	446.559	5.003	5.113	4.168	0.449	0.798	0.765
24MM-C7	165.047	220.278	507.519	5.480	5.752	3.844	0.391	0.636	0.749
2234MMMM-C5	163.603	209.897	410.677	4.731	5.047	4.106	0.463	0.763	0.779
22MM-C7	164.016	221.271	554.019	5.411	5.865	4.168	0.379	0.656	0.741
223MMM-C6	163.938	215.620	456.468	4.911	5.286	4.197	0.431	0.738	0.760
235MMM-C6	163.921	216.403	486.875	5.275	5.384	4.093	0.422	0.745	0.757
244MMM-C6	164.263	214.907	478.821	5.128	5.402	4.127	0.424	0.725	0.764
224MMM-C6	164.000	216.617	471.400	5.085	5.392	4.104	0.421	0.718	0.757
225MMM-C6	164.103	219.201	495.366	5.365	5.269	4.184	0.419	0.766	0.749
2244MMMM-C5	164.104	214.571	408.276	4.639	5.042	4.168	0.455	0.761	0.765

Table 2. Van der Waals Molecular Descriptors of the First 72 Alkanes

Alkane	VW	SW	VEL	EX	EY	EZ	GLOB	GLEL	RWV
C2	45.672	71.059	114.658	3.021	3.145	2.881	0.613	0.880	0.643
C3	62.678	93.293	160.367	3.516	3.780	2.881	0.533	0.709	0.672
C4	79.733	115.287	190.706	3.763	4.200	2.881	0.494	0.615	0.692
2-M-C3	79.699	114.746	217.825	3.889	3.781	3.536	0.536	0.884	0.695
C5	96.825	137.555	247.324	4.252	4.820	2.881	0.438	0.527	0.704
2M-C4	96.719	135.905	264.984	4.182	4.248	3.561	0.503	0.825	0.712
22MM-C3	96.112	135.069	254.783	3.875	3.766	4.168	0.512	0.840	0.712
C6	113.685	159.426	284.774	4.503	5.241	2.881	0.408	0.472	0.713
3M-C5	113.597	156.104	318.161	4.307	4.899	3.599	0.446	0.646	0.728
2M-C5	113.775	157.987	339.431	4.687	4.811	3.594	0.449	0.728	0.720
23MM-C4	113.295	154.842	332.325	4.503	4.220	4.175	0.488	0.869	0.732
22MM-C4	113.230	155.351	311.724	4.215	4.235	4.168	0.516	0.979	0.729
C7	131.027	181.788	352.736	4.988	5.861	2.881	0.369	0.418	0.721
3E-C5	130.432	176.805	390.419	4.661	4.931	4.055	0.449	0.777	0.738
3M-C6	130.657	178.137	369.805	4.503	5.352	3.664	0.411	0.576	0.733
2M-C6	130.675	180.112	389.158	4.940	5.265	3.572	0.413	0.637	0.726
23MM-C5	130.175	174.531	397.190	4.681	4.859	4.168	0.460	0.826	0.746
33MM-C5	130.438	173.979	373.215	4.344	4.921	4.168	0.457	0.748	0.750
223MMM-C4	130.086	173.065	340.481	4.587	4.251	4.169	0.492	0.842	0.752
24-MMC5	130.472	177.729	363.282	4.746	5.027	3.635	0.438	0.683	0.734
22MM-C5	130.431	177.235	390.649	4.663	4.799	4.168	0.460	0.844	0.736
C8	147.814	204.050	397.254	5.240	6.281	2.881	0.346	0.383	0.724
3E-C6	147.572	199.053	423.503	4.662	5.378	4.033	0.414	0.650	0.741
3M-C7	147.843	200.529	459.253	5.026	5.961	3.660	0.371	0.518	0.737
34MM-C6	146.984	194.367	422.458	4.536	5.351	4.156	0.424	0.658	0.756
3E-3M-C5	147.192	193.627	428.593	4.792	4.950	4.314	0.461	0.844	0.760
4M-C7	147.596	200.355	450.614	4.972	5.854	3.696	0.378	0.536	0.737
2M-C7	147.868	201.974	478.805	5.432	5.861	3.591	0.375	0.568	0.732
3E-2M-C5	147.146	193.534	427.292	4.946	4.910	4.201	0.461	0.843	0.760
23MM-C6	147.319	196.670	452.909	4.852	5.351	4.165	0.420	0.706	0.749
233MMM-C5	147.226	191.854	411.251	4.769	4.944	4.164	0.466	0.813	0.767
234MMM-C5	147.420	194.534	397.214	4.780	4.962	3.998	0.458	0.776	0.758
33MM-C6	147.539	195.769	422.747	4.523	5.353	4.168	0.422	0.658	0.754
223MMM-C5	147.123	193.437	401.790	4.740	4.854	4.169	0.470	0.839	0.761
25MM-C6	147.027	200.288	492.840	5.286	5.260	4.232	0.417	0.797	0.734
22MM-C6	147.747	199.191	456.496	4.961	5.270	4.168	0.422	0.744	0.742
2233MMMM-C4	146.856	188.312	338.229	4.563	4.239	4.175	0.513	0.850	0.780
224MMM-C5	147.018	196.564	410.699	4.714	5.036	4.130	0.446	0.768	0.748
C9	164.641	226.192	476.615	5.723	6.901	2.881	0.316	0.346	0.728
33EE-C5	164.061	211.324	459.204	4.769	5.035	4.566	0.463	0.859	0.776
3E-C7	164.668	221.159	532.315	5.028	6.010	4.206	0.372	0.585	0.745
3M-C8	164.821	222.363	518.339	5.259	6.396	3.680	0.348	0.473	0.741
4M-C8	164.737	222.348	518.784	5.214	6.354	3.739	0.350	0.483	0.741
2M-C8	164.651	224.148	535.314	5.679	6.300	3.572	0.350	0.511	0.735
3E-23MM-C5	163.900	210.954	455.680	5.057	5.038	4.269	0.461	0.841	0.777
2334MMMM-C5	163.486	209.776	419.044	4.812	5.003	4.155	0.467	0.799	0.779
4E-C7	164.814	220.815	487.239	5.054	5.713	4.029	0.392	0.624	0.746
3E-3M-C6	164.119	215.689	472.306	4.804	5.410	4.338	0.422	0.712	0.761
23MM-C7	164.329	218.824	558.982	5.408	5.934	4.158	0.380	0.639	0.751
334MMM-C6	163.840	212.499	440.007	4.587	5.505	4.160	0.420	0.629	0.771
2233MMMM-C5	163.704	205.939	400.956	4.676	4.902	4.176	0.486	0.813	0.795
34MM-C7	163.970	215.894	522.987	5.081	5.921	4.150	0.385	0.601	0.759
234MMM-C6	164.105	214.473	460.278	4.964	5.428	4.078	0.423	0.687	0.765
233MMM-C6	164.180	213.892	455.137	4.831	5.398	4.167	0.427	0.691	0.768
33MM-C7	164.007	217.971	529.200	5.085	5.962	4.168	0.379	0.596	0.752
3E-24MM-C5	164.363	216.103	427.188	5.057	5.064	3.982	0.451	0.785	0.761
35MM-C7	164.491	218.873	533.826	5.433	5.307	4.420	0.415	0.795	0.752
25MM-C7	164.140	220.135	568.077	5.393	5.886	4.272	0.380	0.665	0.746
26MM-C7	164.715	222.197	491.514	5.443	5.995	3.596	0.371	0.545	0.741
44MM-C7	164.587	217.682	501.828	4.898	5.869	4.168	0.386	0.593	0.756
4E-2M-C6	164.549	219.575	436.658	5.125	5.414	3.757	0.415	0.657	0.749
3E-22MM-C5	164.151	214.491	446.559	5.003	5.113	4.168	0.449	0.798	0.765
24MM-C7	165.047	220.278	507.519	5.480	5.752	3.844	0.391	0.636	0.749
2234MMMM-C5	163.603	209.897	410.677	4.731	5.047	4.106	0.463	0.763	0.779
22MM-C7	164.016	221.271	554.019	5.411	5.865	4.168	0.379	0.656	0.741
223MMM-C6	163.938	215.620	456.468	4.911	5.286	4.197	0.431	0.738	0.760
235MMM-C6	163.921	216.403	486.875	5.275	5.384	4.093	0.422	0.745	0.757
244MMM-C6	164.263	214.907	478.821	5.128	5.402	4.127	0.424	0.725	0.764
224MMM-C6	164.000	216.617	471.400	5.085	5.392	4.104	0.421	0.718	0.757
225MMM-C6	164.103	219.201	495.366	5.365	5.269	4.184	0.419	0.766	0.749
2244MMMM-C5	164.104	214.571	408.276	4.639	5.042	4.168	0.455	0.761	0.765

Intercorrelation of Topological Distance Indices and van Der Waals Molecular Descriptors

In this section we analyze the extent to which the molecular descriptors presented in this paper are linearly intercorrelated. The correlation analysis was performed on all TDIs and vdWMDs considered in this report for a set of 72 alkanes of up to 9 carbon atoms. For this purpose alkanes are convenient systems because they represent structurally rather simple chemical structures, and skeletal branching is their only complicated structural feature [21]. In this way we can establish to what extent the molecular descriptors from Table 1a and 1 are orthogonal. This orthogonality is absolutely necessary for molecular descriptors in QSPR relations because it avoids the artificial strengthening of correlations. It also assures that a quantity of information is independent of the parameters of the obtained linear model, thus very useful for physical interpretations of the model. If, on the other hand, the MDs are not orthogonal, it is possible that they predominantly express the same type of structural information, with differences residing in the scaling factors.

We have investigated the linear relationship between the pairs of molecular descriptors presented here, MD_a and MD_b,

MD_a = α + β∙MD_b

(34)

where MD_a and MD_b are TDIs, GTDIs and vdwMDs.

The correlation coefficient, r, is a measure of linear relationship described in relation (34). If r = 0 no linear relationship exists between MD_a and MD_b. If r = 1, there is a direct linear relationship, and if r = -1 , there is an inverse linear relationship between MD_a and MD_b. The correlation coefficient r ≥ 0.900 was proposed as the criterion for the intercorrelated pairs of molecular descriptors [48]. Strongly intercorrelated pairs of the molecular descriptors are those with r ≥ 0.980.

The results of the correlation analysis are displayed as the intercorrelation matrices with the correlation coefficient r. In Table 3a, Table 3b, Table 3c and Table 3d we give the intercorrelation matrices reflecting pairwise linear correlation for all molecular descriptors from Table 1 and Table 2: 11 selected TDIs, 16 GTDIs extended here from the reciprocical distance matrix, and 9 vdWMDs. The Table 3a, Table 3b and Table 3c contain the intercorrelation matrices corresponding to TDIs, GTDIs and vdWMDs, respectively. Since the matrices are symmetric, we give only the upper triangle. In Table 3d we report the intercorrelation matrix of TDIs and GTDIs.

Table 3a. Intercorrelation Matrix of Topological Distance Indices for Alkanes with up to 9 Carbon Atoms

	W	P	F	J	VAD1	VAD2	VAD3	VED1	VED2	VED3	VRD
W	1.000	0.719	0.716	0.523	0.945	0.860	0.862	0.915	-0.810	0.874	0.952
P		1.000	0.842	0.850	0.825	0.766	0.784	0.821	-0.744	0.793	0.838
F			1.000	0.933	0.757	0.666	0.802	0.861	-0.805	0.842	0.873
J				1.000	0.113	0.094	0.278	0.281	-0.344	0.312	0.241
VAD1					1.000	0.968	0.918	0.934	-0.854	0.905	0.933
VAD2						1.000	0.927	0.897	-0.866	0.890	0.849
VAD3							1.000	0.982	-0.989	0.993	0.927
VED1								1.000	-0.967	0.993	0.978
VED2									1.000	-0.990	-0.901
VED3										1.000	0.950
VRD											1.000

Table 3a. Intercorrelation Matrix of Topological Distance Indices for Alkanes with up to 9 Carbon Atoms

	W	P	F	J	VAD1	VAD2	VAD3	VED1	VED2	VED3	VRD
W	1.000	0.719	0.716	0.523	0.945	0.860	0.862	0.915	-0.810	0.874	0.952
P		1.000	0.842	0.850	0.825	0.766	0.784	0.821	-0.744	0.793	0.838
F			1.000	0.933	0.757	0.666	0.802	0.861	-0.805	0.842	0.873
J				1.000	0.113	0.094	0.278	0.281	-0.344	0.312	0.241
VAD1					1.000	0.968	0.918	0.934	-0.854	0.905	0.933
VAD2						1.000	0.927	0.897	-0.866	0.890	0.849
VAD3							1.000	0.982	-0.989	0.993	0.927
VED1								1.000	-0.967	0.993	0.978
VED2									1.000	-0.990	-0.901
VED3										1.000	0.950
VRD											1.000

From Table 3 we learn several interesting points:

1)

The intercorrelation matrix of the selected topological indices presented in Table 3a reveals that these indices are not strongly intercorrelated, that is their information content about topological structure of the 72 alkanes from table 1 is somewhat independent. Only the indices derived from eigenvalues and eigenvectors are better intercorrelated. The TDIs belonging to this group are very poorly correlated with Balaban’s J-index. Besides, J is also independent when compared to W, and very weakly linked to P. On the other hand it seems to correlate very well with F (r = 0.933). From this point of view it is necessary to avoid the simultaneous use of these indices for studying physical properties in QSPR relations.

2)

The majority of GTDIs, ^kδ_λ (k = 1,2,3,4; λ = 0,1,2,3) counterparts are strongly intercorrelated. Taking as criterion for strong correlations r ≥ 0.980 one notices that there exists a strong correlation inside each class λ, which slightly decreases along with the increase in k. This fact is entirely explainable, if we take into consideration the way in which LOVIs are constructed; the more the dimensionality of the space is increased, the interaction between atoms that are separated by the same topological distance decreases, and the influence gets smaller as the distance and the dimensionality of the space get larger. The degree of correlation between indices ^kδ_λ of different classes are generally smaller, except those corresponding to λ = 1 and λ = 2, which are greater than r = 0.960.

Table 3b. Intercorrelation Matrix of Generalized Topological Distance Indices for Alkanes with up to 9 Carbon Atoms

	1δο	2δο	3δο	4δο	1δ1	2δ1	3δ1	4δ1	1δ2	2δ2	3δ2	4δ2	1δ3	2δ3	3δ3	4δ3
1δο	1.000	0.999	0.996	0.993	0.965	0.978	0.990	0.994	0.809	0.859	0.899	0.922	0.837	0.939	0.967	0.972
2δο		1.000	0.999	0.996	0.967	0.980	0.992	0.997	0.810	0.858	0.898	0.921	0.855	0.948	0.973	0.979
3δο			1.000	0.999	0.978	0.988	0.997	1.000	0.835	0.879	0.915	0.936	0.868	0.959	0.979	0.981
4δο				1.000	0.986	0.994	0.999	1.000	0.856	0.897	0.930	0.948	0.872	0.965	0.981	0.979
1δ1					1.000	0.998	0.991	0.982	0.931	0.958	0.976	0.985	0.861	0.967	0.967	0.952
2δ1						1.000	0.997	0.991	0.908	0.941	0.964	0.977	0.862	0.967	0.973	0.962
3δ1							1.000	0.999	0.873	0.912	0.942	0.959	0.867	0.965	0.979	0.974
4δ1								1.000	0.845	0.887	0.922	0.941	0.872	0.963	0.981	0.981
1δ2									1.000	0.994	0.980	0.967	0.745	0.872	0.837	0.795
2δ2										1.000	0.996	0.988	0.754	0.891	0.867	0.831
3δ2											1.000	0.998	0.769	0.907	0.893	0.864
4δ2												1.000	0.780	0.918	0.910	0.885
1δ3													1.000	0.959	0.944	0.937
2δ3														1.000	0.994	0.982
3δ3															1.000	0.997
4δ3																1.000

Table 3b. Intercorrelation Matrix of Generalized Topological Distance Indices for Alkanes with up to 9 Carbon Atoms

	1δο	2δο	3δο	4δο	1δ1	2δ1	3δ1	4δ1	1δ2	2δ2	3δ2	4δ2	1δ3	2δ3	3δ3	4δ3
1δο	1.000	0.999	0.996	0.993	0.965	0.978	0.990	0.994	0.809	0.859	0.899	0.922	0.837	0.939	0.967	0.972
2δο		1.000	0.999	0.996	0.967	0.980	0.992	0.997	0.810	0.858	0.898	0.921	0.855	0.948	0.973	0.979
3δο			1.000	0.999	0.978	0.988	0.997	1.000	0.835	0.879	0.915	0.936	0.868	0.959	0.979	0.981
4δο				1.000	0.986	0.994	0.999	1.000	0.856	0.897	0.930	0.948	0.872	0.965	0.981	0.979
1δ1					1.000	0.998	0.991	0.982	0.931	0.958	0.976	0.985	0.861	0.967	0.967	0.952
2δ1						1.000	0.997	0.991	0.908	0.941	0.964	0.977	0.862	0.967	0.973	0.962
3δ1							1.000	0.999	0.873	0.912	0.942	0.959	0.867	0.965	0.979	0.974
4δ1								1.000	0.845	0.887	0.922	0.941	0.872	0.963	0.981	0.981
1δ2									1.000	0.994	0.980	0.967	0.745	0.872	0.837	0.795
2δ2										1.000	0.996	0.988	0.754	0.891	0.867	0.831
3δ2											1.000	0.998	0.769	0.907	0.893	0.864
4δ2												1.000	0.780	0.918	0.910	0.885
1δ3													1.000	0.959	0.944	0.937
2δ3														1.000	0.994	0.982
3δ3															1.000	0.997
4δ3																1.000

Table 3c. Intercorrelation Matrix of van der Waals Molecular Descriptors for Alkanes with up to 9 Carbon Atoms

	VW	SW	VEL	EX	EY	EZ	GLOB	GLEL	RWV
VW	1.000	0.994	0.924	0.852	0.751	0.566	-0.661	-0.220	0.839
SW		1.000	0.944	0.891	0.806	0.511	-0.729	-0.294	0.782
VEL			1.000	0.906	0.822	0.538	-0.764	-0.288	0.652
EX				1.000	0.850	0.270	-0.825	-0.413	0.528
EY					1.000	0.018	-0.978	-0.761	0.351
EZ						1.000	0.056	0.564	0.757
GLOB							1.000	0.812	-0.240
GLEL								1.000	0.178
RWV									1.000

Table 3c. Intercorrelation Matrix of van der Waals Molecular Descriptors for Alkanes with up to 9 Carbon Atoms

	VW	SW	VEL	EX	EY	EZ	GLOB	GLEL	RWV
VW	1.000	0.994	0.924	0.852	0.751	0.566	-0.661	-0.220	0.839
SW		1.000	0.944	0.891	0.806	0.511	-0.729	-0.294	0.782
VEL			1.000	0.906	0.822	0.538	-0.764	-0.288	0.652
EX				1.000	0.850	0.270	-0.825	-0.413	0.528
EY					1.000	0.018	-0.978	-0.761	0.351
EZ						1.000	0.056	0.564	0.757
GLOB							1.000	0.812	-0.240
GLEL								1.000	0.178
RWV									1.000

Table 3d. Intercorrelation Matrix of Generalized Topological Distance Indices (GTDIs) against Topological Distance Indices (TDIs) for Alkanes with up to 9 Carbon Atoms

	W	P	F	J	VAD1	VAD2	VAD3	VED1	VED2	VED3	VRD
1δο	0.923	0.885	0.914	0.801	0.937	0.857	0.925	0.975	-0.896	0.947	0.991
2δο	0.912	0.881	0.920	0.817	0.932	0.861	0.939	0.982	-0.916	0.960	0.989
3δο	0.921	0.866	0.905	0.799	0.938	0.874	0.952	0.990	-0.930	0.971	0.991
4δο	0.931	0.852	0.888	0.778	0.944	0.884	0.959	0.994	-0.936	0.975	0.992
1δ1	0.959	0.789	0.802	0.672	0.955	0.910	0.965	0.989	-0.936	0.973	0.981
2δ1	0.954	0.817	0.832	0.710	0.955	0.903	0.964	0.993	-0.937	0.975	0.988
3δ1	0.939	0.844	0.871	0.758	0.948	0.890	0.961	0.995	-0.937	0.976	0.993
4δ1	0.925	0.857	0.897	0.789	0.939	0.876	0.956	0.992	-0.934	0.974	0.992
1δ2	0.924	0.582	0.534	0.375	0.887	0.882	0.882	0.880	-0.842	0.870	0.858
2δ2	0.947	0.660	0.598	0.452	0.922	0.904	0.907	0.913	-0.865	0.898	0.899
3δ2	0.958	0.724	0.659	0.526	0.945	0.918	0.929	0.940	-0.887	0.924	0.930
4δ2	0.960	0.760	0.698	0.573	0.955	0.923	0.941	0.956	-0.900	0.938	0.947
1δ3	0.780	0.555	0.824	0.696	0.752	0.719	0.880	0.891	-0.905	0.904	0.850
2δ3	0.916	0.694	0.844	0.694	0.888	0.837	0.945	0.972	-0.939	0.965	0.956
3δ3	0.912	0.755	0.894	0.756	0.897	0.833	0.942	0.979	-0.934	0.966	0.973
4δ3	0.892	0.780	0.927	0.797	0.885	0.813	0.930	0.972	-0.923	0.958	0.970

Table 3d. Intercorrelation Matrix of Generalized Topological Distance Indices (GTDIs) against Topological Distance Indices (TDIs) for Alkanes with up to 9 Carbon Atoms

	W	P	F	J	VAD1	VAD2	VAD3	VED1	VED2	VED3	VRD
1δο	0.923	0.885	0.914	0.801	0.937	0.857	0.925	0.975	-0.896	0.947	0.991
2δο	0.912	0.881	0.920	0.817	0.932	0.861	0.939	0.982	-0.916	0.960	0.989
3δο	0.921	0.866	0.905	0.799	0.938	0.874	0.952	0.990	-0.930	0.971	0.991
4δο	0.931	0.852	0.888	0.778	0.944	0.884	0.959	0.994	-0.936	0.975	0.992
1δ1	0.959	0.789	0.802	0.672	0.955	0.910	0.965	0.989	-0.936	0.973	0.981
2δ1	0.954	0.817	0.832	0.710	0.955	0.903	0.964	0.993	-0.937	0.975	0.988
3δ1	0.939	0.844	0.871	0.758	0.948	0.890	0.961	0.995	-0.937	0.976	0.993
4δ1	0.925	0.857	0.897	0.789	0.939	0.876	0.956	0.992	-0.934	0.974	0.992
1δ2	0.924	0.582	0.534	0.375	0.887	0.882	0.882	0.880	-0.842	0.870	0.858
2δ2	0.947	0.660	0.598	0.452	0.922	0.904	0.907	0.913	-0.865	0.898	0.899
3δ2	0.958	0.724	0.659	0.526	0.945	0.918	0.929	0.940	-0.887	0.924	0.930
4δ2	0.960	0.760	0.698	0.573	0.955	0.923	0.941	0.956	-0.900	0.938	0.947
1δ3	0.780	0.555	0.824	0.696	0.752	0.719	0.880	0.891	-0.905	0.904	0.850
2δ3	0.916	0.694	0.844	0.694	0.888	0.837	0.945	0.972	-0.939	0.965	0.956
3δ3	0.912	0.755	0.894	0.756	0.897	0.833	0.942	0.979	-0.934	0.966	0.973
4δ3	0.892	0.780	0.927	0.797	0.885	0.813	0.930	0.972	-0.923	0.958	0.970

3)

Van der Waals molecular descriptors, vdWMDs, are much more independent relative to each other than the GTDIs and TDIs. A strong correlation was observed only between the volume (V^W) and the corresponding vdW surface (S^W) of the 72 alkanes having 2 – 9 carbon atoms (r = 0.994). This significant correlation was obtained between the vdW volume and surface, but also between them and the molecular vdW volume of alkanes treated as molecules with a more or less ellipsoidal shape. The shift of alkanes to an extended, intercalated conformation greatly influences the volume of the ellipsoid and progressively smaller the vdW surface area and the vdW volume. On the other hand, conformational variations on orthogonal directions are affecting these descriptors on a much smaller measure. Our intercorrelation results suggest the possibility of simultaneously using these indices in QSAR and QSPR relations for global testing of vdW space occupied by molecules (space-filling), along with bulk steric parameters (V^W, S^W, V^EL, G^LOB, G^LEL, R^WV), or certain directions within them (E_X, E_Y, E_Z). The simple and fast calculus for any molecular structure and the possibility of immediately testing the degree of orthogonality ensures their large applicability for any series of compounds.

4)

Generalized topological distance indices derived from the reciprocical distance matrix, GTDIs, present significant correlations with topological indices derived from eigenvalues and eigenvectors of the distance matrix, D. Repeatedly, the strongest are those between ^kδ_λ (k = 1,2,3,4, λ = 0,1) and VRD. In this case a more rigorous statistical analysis is imposed on the relation between distance indices, ^kδ_λ, and the VADi and VEDi parameters, i = 1,2,3. The intercorrelation between GTDIs and the first indices defined on the distance matrix is decreasing in the following order: W, F, P. Although, generally speaking, the Wiener index, W, correlates well with GTDIs, there are two surprising exceptions for indices ¹δ₃ and ⁴δ₃. Investigating the physical meaning of GTDIs could emerge interesting information on other topological indices. The work is in progress.

5)

Are the topological indices steric measures of molecular van der Waals space? Although some reported that they correlate well with molecular volume [7] or surface area, extensive studies on this subject have not yet been performed. In Table 4 we present the intercorrelation matrix of molecular vdW descriptors and of topological indices described in this work. The best results were obtained for the correlations with the van der Waals molecular volume (V^W) and surface (S^W) against Wiener indices (W), derived from eigenvectors and eigenvalues of the distance matrix, and GTDIs, ^kδ_λ, except for indices with λ = 2 and k = 1 (r = 0.886), and λ = 3 and k = 1 (r = 0.869). Except for P, F and J indices, the others should be viewed as bulk steric parameters, as measured by vdW volume and surface of tested alkanes. The steric component of most topological indices is poorly explained by vdW volumes of ellipsoid-assimilated alkanes (revolving around r = 0.900). Weak correlations were also obtained for P, F and J. The results suggest the impossibility of testing the vector nature of steric effects by means of topological distance indices, which is rather important for modeling biological interactions. This is a possible explanation for the lesser utility of topological indices for QSAR studies.

Table 4. Intercorrelation Matrix of All Topological Distance Indices (TDIs and GTDIs) against van der Walls Molecular Descriptors (vdWMDs)

	VW	SW	VEL	EX	EY	EZ	GLOB	GLEL	RWV
W	0.944	0.958	0.930	0.873	0.830	0.386	-0.741	-0.373	0.642
P	0.834	0.778	0.666	0.519	0.416	0.636	-0.274	0.074	0.913
F	0.859	0.809	0.687	0.559	0.354	0.751	-0.237	0.191	0.937
J	0.743	0.677	0.537	0.392	0.179	0.820	-0.073	0.343	0.970
VAD1	0.951	0.950	0.895	0.831	0.779	0.455	-0.680	-0.295	0.747
VAD2	0.896	0.902	0.847	0.826	0.795	0.386	-0.722	-0.358	0.711
VAD3	0.965	0.965	0.890	0.857	0.761	0.540	-0.702	-0.258	0.840
VED1	0.996	0.990	0.916	0.854	0.744	0.581	-0.664	-0.211	0.856
VED2	-0.940	-0.939	-0.860	-0.832	-0.717	-0.570	0.670	0.215	-0.849
VED3	0.978	0.975	0.898	0.851	0.738	0.581	-0.672	-0.213	0.860
VRD	0.991	0.981	0.910	0.820	0.718	0.572	-0.618	-0.191	0.829
1δο	0.986	0.965	0.880	0.776	0.656	0.621	-0.546	-0.112	0.878
2δο	0.989	0.968	0.881	0.782	0.656	0.632	-0.550	-0.107	0.892
3δο	0.995	0.979	0.897	0.808	0.687	0.616	-0.587	-0.142	0.880
4δο	0.998	0.986	0.909	0.827	0.714	0.597	-0.618	-0.173	0.865
1δ1	0.994	0.999	0.944	0.890	0.812	0.504	-0.733	-0.303	0.783
2δ1	0.999	0.998	0.936	0.869	0.779	0.541	-0.694	-0.256	0.813
3δ1	1.000	0.991	0.920	0.841	0.732	0.585	-0.640	-0.194	0.850
4δ1	0.997	0.983	0.905	0.820	0.698	0.612	-0.601	-0.151	0.873
1δ2	0.886	0.926	0.916	0.932	0.951	0.229	-0.913	-0.574	0.532
2δ2	0.922	0.953	0.934	0.919	0.925	0.305	-0.872	-0.510	0.600
3δ2	0.950	0.971	0.941	0.905	0.893	0.373	-0.828	-0.445	0.665
4δ2	0.965	0.980	0.943	0.895	0.869	0.416	-0.797	-0.402	0.706
1δ3	0.869	0.876	0.809	0.808	0.633	0.542	-0.598	-0.151	0.738
2δ3	0.968	0.975	0.913	0.881	0.752	0.533	-0.689	-0.239	0.775
3δ3	0.978	0.974	0.900	0.843	0.699	0.589	-0.621	-0.165	0.827
4δ3	0.972	0.959	0.875	0.804	0.647	0.626	-0.561	-0.103	0.858

Table 4. Intercorrelation Matrix of All Topological Distance Indices (TDIs and GTDIs) against van der Walls Molecular Descriptors (vdWMDs)

	VW	SW	VEL	EX	EY	EZ	GLOB	GLEL	RWV
W	0.944	0.958	0.930	0.873	0.830	0.386	-0.741	-0.373	0.642
P	0.834	0.778	0.666	0.519	0.416	0.636	-0.274	0.074	0.913
F	0.859	0.809	0.687	0.559	0.354	0.751	-0.237	0.191	0.937
J	0.743	0.677	0.537	0.392	0.179	0.820	-0.073	0.343	0.970
VAD1	0.951	0.950	0.895	0.831	0.779	0.455	-0.680	-0.295	0.747
VAD2	0.896	0.902	0.847	0.826	0.795	0.386	-0.722	-0.358	0.711
VAD3	0.965	0.965	0.890	0.857	0.761	0.540	-0.702	-0.258	0.840
VED1	0.996	0.990	0.916	0.854	0.744	0.581	-0.664	-0.211	0.856
VED2	-0.940	-0.939	-0.860	-0.832	-0.717	-0.570	0.670	0.215	-0.849
VED3	0.978	0.975	0.898	0.851	0.738	0.581	-0.672	-0.213	0.860
VRD	0.991	0.981	0.910	0.820	0.718	0.572	-0.618	-0.191	0.829
1δο	0.986	0.965	0.880	0.776	0.656	0.621	-0.546	-0.112	0.878
2δο	0.989	0.968	0.881	0.782	0.656	0.632	-0.550	-0.107	0.892
3δο	0.995	0.979	0.897	0.808	0.687	0.616	-0.587	-0.142	0.880
4δο	0.998	0.986	0.909	0.827	0.714	0.597	-0.618	-0.173	0.865
1δ1	0.994	0.999	0.944	0.890	0.812	0.504	-0.733	-0.303	0.783
2δ1	0.999	0.998	0.936	0.869	0.779	0.541	-0.694	-0.256	0.813
3δ1	1.000	0.991	0.920	0.841	0.732	0.585	-0.640	-0.194	0.850
4δ1	0.997	0.983	0.905	0.820	0.698	0.612	-0.601	-0.151	0.873
1δ2	0.886	0.926	0.916	0.932	0.951	0.229	-0.913	-0.574	0.532
2δ2	0.922	0.953	0.934	0.919	0.925	0.305	-0.872	-0.510	0.600
3δ2	0.950	0.971	0.941	0.905	0.893	0.373	-0.828	-0.445	0.665
4δ2	0.965	0.980	0.943	0.895	0.869	0.416	-0.797	-0.402	0.706
1δ3	0.869	0.876	0.809	0.808	0.633	0.542	-0.598	-0.151	0.738
2δ3	0.968	0.975	0.913	0.881	0.752	0.533	-0.689	-0.239	0.775
3δ3	0.978	0.974	0.900	0.843	0.699	0.589	-0.621	-0.165	0.827
4δ3	0.972	0.959	0.875	0.804	0.647	0.626	-0.561	-0.103	0.858

Correlations with Boiling Points of Alkanes

In order to check whether the selected topological indices, generalized topological distance indices and van der Waals molecular descriptors can be used in correlations with experimentally measured properties (in QSPR) we focused on boiling points. The choice was motivated by the fact that boiling points are known to depend on molecular constitution (graph topology). The true nature of the intermolecular forces involved and the entropy change in the transition from liquid to gas phase are not considered in detail. We are interested here to test the correlation ability of the considered molecular descriptors, topological indices and van der Waals structural indices.

Monoparametric correlations with boiling points (at normal pressure) for all 72 alkanes with N = 2 – 9 carbon atoms were tested for the 36 described molecular descriptors (MD) following a linear equation,

BP = α∙(±Δα)+ β∙(±Δβ)∙MD

(35)

where MD = TDIs, GTDIs and vdWMDs, and statistical characteristics of each correlation were considered (see Table 4). In Table 5 r is the correlation coefficient, s is the standard deviation, EV is the explained variance, t is the Student test for r and F is the Fisher test for 72 degrees of freedom.

Table 5. Statistical Characteristics of Structure – Boiling Point Models (35) with 11 Selected Topological Distance Indices, 16 Generalized Topological Distance Indices and 9 van der Waals Molecular Descriptors

Eq.	Xi	r	α	±Δα	β	±Δβ	s	F	EV	t
36	W	0.916	4.31	5.70	1.42	0.07	18.715	374.5	0.837	19.4
37	F	0.834	19.13	7.43	13.18	1.03	25.728	164.3	0.691	12.8
38	P	0.803	-8.29	10.53	6.99	0.61	27.783	130.6	0.640	11.4
39	J	0.722	-85.26	21.97	60.77	6.87	32.260	78.3	0.514	8.8
40	VAD1	0.947	-29.08	5.66	7.57	0.30	14.910	631.5	0.896	25.1
41	VAD2	0.925	-103.00	10.34	95.04	4.60	17.714	426.4	0.854	20.6
42	VAD3	0.984	-36.40	3.18	56.68	1.20	8.259	2221.0	0.968	47.1
43	VED1	0.989	-294.47	6.95	146.42	2.52	6.744	3366.1	0.979	58.0
44	VED2	0.960	370.10	9.15	-731.86	25.02	12.985	855.6	0.921	29.3
45	VED3	0.984	24.68	1.96	112.36	2.36	8.177	2267.0	0.969	47.6
46	VRD	0.962	-44.30	5.28	6.82	0.23	12.802	882.2	0.924	29.7
47	1δο	0.956	-43.28	5.66	5.12	0.19	13.716	759.3	0.912	27.6
48	2δο	0.963	-63.35	5.82	8.51	0.28	12.635	907.7	0.926	30.1
49	3δο	0.973	-77.71	5.31	11.24	0.32	10.786	1272.2	0.946	35.7
50	4δο	0.979	-85.16	4.79	12.85	0.31	9.439	1683.2	0.958	41.0
51	1δ1	0.988	-218.36	6.16	78.24	1.47	7.341	2830.3	0.975	53.2
52	2δ1	0.987	-166.85	5.25	53.23	1.01	7.403	2781.4	0.974	52.7
53	3δ1	0.983	-146.99	5.74	43.90	0.98	8.675	2006.0	0.965	44.8
54	4δ1	0.975	-137.52	6.59	39.85	1.06	10.260	1413.6	0.951	37.6
55	1δ2	0.911	-235.00	18.32	97.53	5.20	19.205	352.0	0.828	18.8
56	2δ2	0.941	-140.75	10.66	49.69	2.11	15.817	553.1	0.883	23.5
57	3δ2	0.963	-122.07	7.70	38.08	1.26	12.603	912.7	0.926	30.2
58	4δ2	0.974	-117.21	6.23	34.00	0.93	10.534	1337.3	0.948	36.6
59	1δ3	0.832	-299.41	32.01	52.84	4.15	25.849	162.1	0.688	12.7
60	2δ3	0.941	-158.93	11.41	20.34	0.86	15.800	554.5	0.884	23.5
61	3δ3	0.945	-118.22	9.37	12.88	0.53	15.304	595.7	0.891	24.4
62	4δ3	0.932	-100.17	9.65	10.23	0.47	16.877	477.0	0.867	21.8
63	VW	0.986	-140.62	5.06	1.71	0.03	7.852	2464.6	0.971	49.6
64	SW	0.984	-165.00	5.89	1.40	0.03	8.339	2177.0	0.968	46.7
65	VEL	0.911	-83.11	10.35	0.45	0.02	19.229	351.0	0.827	18.7
66	EX	0.849	-291.59	29.28	82.60	6.05	24.600	186.4	0.718	13.7
67	EY	0.790	-178.72	26.28	54.54	4.99	28.575	119.5	0.619	10.9
68	EZ	0.523	-112.71	42.28	55.91	10.73	39.718	27.1	0.264	5.2
69	GLOB	0.710	383.22	32.62	-642.13	75.10	32.829	73.1	0.497	8.6
70	GLEL	0.285	176.94	28.50	-101.46	40.22	44.673	6.4	0.069	2.5
71	RWV	0.833	-1060.80	91.34	1568.38	122.68	25.773	163.4	0.690	12.8

Table 5. Statistical Characteristics of Structure – Boiling Point Models (35) with 11 Selected Topological Distance Indices, 16 Generalized Topological Distance Indices and 9 van der Waals Molecular Descriptors

Eq.	Xi	r	α	±Δα	β	±Δβ	s	F	EV	t
36	W	0.916	4.31	5.70	1.42	0.07	18.715	374.5	0.837	19.4
37	F	0.834	19.13	7.43	13.18	1.03	25.728	164.3	0.691	12.8
38	P	0.803	-8.29	10.53	6.99	0.61	27.783	130.6	0.640	11.4
39	J	0.722	-85.26	21.97	60.77	6.87	32.260	78.3	0.514	8.8
40	VAD1	0.947	-29.08	5.66	7.57	0.30	14.910	631.5	0.896	25.1
41	VAD2	0.925	-103.00	10.34	95.04	4.60	17.714	426.4	0.854	20.6
42	VAD3	0.984	-36.40	3.18	56.68	1.20	8.259	2221.0	0.968	47.1
43	VED1	0.989	-294.47	6.95	146.42	2.52	6.744	3366.1	0.979	58.0
44	VED2	0.960	370.10	9.15	-731.86	25.02	12.985	855.6	0.921	29.3
45	VED3	0.984	24.68	1.96	112.36	2.36	8.177	2267.0	0.969	47.6
46	VRD	0.962	-44.30	5.28	6.82	0.23	12.802	882.2	0.924	29.7
47	1δο	0.956	-43.28	5.66	5.12	0.19	13.716	759.3	0.912	27.6
48	2δο	0.963	-63.35	5.82	8.51	0.28	12.635	907.7	0.926	30.1
49	3δο	0.973	-77.71	5.31	11.24	0.32	10.786	1272.2	0.946	35.7
50	4δο	0.979	-85.16	4.79	12.85	0.31	9.439	1683.2	0.958	41.0
51	1δ1	0.988	-218.36	6.16	78.24	1.47	7.341	2830.3	0.975	53.2
52	2δ1	0.987	-166.85	5.25	53.23	1.01	7.403	2781.4	0.974	52.7
53	3δ1	0.983	-146.99	5.74	43.90	0.98	8.675	2006.0	0.965	44.8
54	4δ1	0.975	-137.52	6.59	39.85	1.06	10.260	1413.6	0.951	37.6
55	1δ2	0.911	-235.00	18.32	97.53	5.20	19.205	352.0	0.828	18.8
56	2δ2	0.941	-140.75	10.66	49.69	2.11	15.817	553.1	0.883	23.5
57	3δ2	0.963	-122.07	7.70	38.08	1.26	12.603	912.7	0.926	30.2
58	4δ2	0.974	-117.21	6.23	34.00	0.93	10.534	1337.3	0.948	36.6
59	1δ3	0.832	-299.41	32.01	52.84	4.15	25.849	162.1	0.688	12.7
60	2δ3	0.941	-158.93	11.41	20.34	0.86	15.800	554.5	0.884	23.5
61	3δ3	0.945	-118.22	9.37	12.88	0.53	15.304	595.7	0.891	24.4
62	4δ3	0.932	-100.17	9.65	10.23	0.47	16.877	477.0	0.867	21.8
63	VW	0.986	-140.62	5.06	1.71	0.03	7.852	2464.6	0.971	49.6
64	SW	0.984	-165.00	5.89	1.40	0.03	8.339	2177.0	0.968	46.7
65	VEL	0.911	-83.11	10.35	0.45	0.02	19.229	351.0	0.827	18.7
66	EX	0.849	-291.59	29.28	82.60	6.05	24.600	186.4	0.718	13.7
67	EY	0.790	-178.72	26.28	54.54	4.99	28.575	119.5	0.619	10.9
68	EZ	0.523	-112.71	42.28	55.91	10.73	39.718	27.1	0.264	5.2
69	GLOB	0.710	383.22	32.62	-642.13	75.10	32.829	73.1	0.497	8.6
70	GLEL	0.285	176.94	28.50	-101.46	40.22	44.673	6.4	0.069	2.5
71	RWV	0.833	-1060.80	91.34	1568.38	122.68	25.773	163.4	0.690	12.8

It can be seen from Table 5 that the correlation coefficients are satisfactory for the majority of generalized topological distance indices (except ¹δ₃) and eigenvalues and eigenvectors based indices VxDn (x = A, E; n = 1 – 3), and unsatisfactory for P, F and especially J topological distance indices and van der Waals molecular descriptors that measure globularity (G^LOB, G^LEL and R^WV) and various directions in molecular van der Waals space of alkanes (E_X, E_Y and E_Z). The best results are obtained for VED1, VED3, ¹δ₁, ²δ₁, ³δ₁, V^W and S^W (r > 0.980). These topological indices contain in a great measure a bulky component and there is a strong relation between them and the whole space of alkane molecules. Van der Waals volume and surface seem to be essential for explaining the structural variation of the boiling points of alkanes. This is easy to explain if we consider the nature of the physical interactions which appear between molecules in the liquid phase and in the gas phase.

The r values for E_X, E_Y, G^LOB and R^WV are lower than those obtained for V^W, S^W and V^EL. The correlation coefficient for P and F topological indices are also fairly low. Poor values for r are especially observed for G^LEL and E_Z, although there is no strong linear relation between them (the coefficient of intercorrelation is r = 0.761 – see Table 3c). This fact demonstrates that these indices contain little (E_X, E_Y, G^LOB and R^WV) or no information (J, E_Z, and G^LEL) about the size of alkane molecules.

The most accurate models are (42), (43), (45), (51) – (53), (63) and (64), where r > 0.980, F are in the intervals 2000 – 3366 (for VED1) and standard deviations vary from 6.74 to 8.66, that is they are less than 3.6% from the whole domain of boiling points. The correlation equations above explain more than 96% from the variance of the experimentally measured boiling points.

The topological distance indices W, P, F and J, and also the van der Waals molecular descriptors (E_X, E_Y), G^LOB and R^WV are less successful in modeling boiling points than the generalized and eigenvalues/eigenvectors distance indices. The worst correlation was obtained with G^LEL, probably because this globularity vdW descriptor, which contains information about the shape of molecules, is normalized; its value tends towards 1 when the shape of the molecule gets closer to a sphere [49].

The results here obtained suggest that the shape of the molecules seems to be less important than the size for structure-based modeling of boiling points of alkanes. Obviously, the shape is also a more abstract concept than size, thus it is also more difficult to estimate it quantitatively (through a single number) than size.

Concluding remarks

In this work we carried out a comparative study of 36 molecular descriptors derived from the distance matrix and molecular van der Waals space. This study belongs to our interest to develop molecular descriptors with clear physical meaning. Among the studied indices were 9 van der Waals descriptors of molecular size and shape and 27 topological distance indices. We also introduced here a generalized formula for deriving topological indices from the reciprocical distance matrix.

We analyzed the correlation among some classical distance matrices (W, P, F and J), distance indices derived by us from eigenvalues and eigenvectors of distance matrix and generalized distance indices, ^kδ_λ (k = 1 – 4, λ = 1 – 3), and their relation with a series of van der Waals molecular descriptors of molecular size (volume and surface area) and shape (globularity measures and ellipsoidal characteristics). The analysis of the link between the topological molecular space and the molecular van der Waals space allowed some insights on the physical meaning of topological indices. The obtained results suggest the possibility of considering topological distance indices as descriptors of the molecule’s size. They can be regarded as bulk parameters. In the series under study the informational content is progressively decreasing in the following approximate order: VRD, ^kδ_λ, VxDn, W, F, P, J.

The correlation analysis on the first 72 alkanes revealed that many counterparts are characterized by their topological distance indices. The meaning of this result is that the topological distance indices contain similar structural information about the molecular graph. On the contrary, the intercorrelation analysis of van der Waals molecular descriptors shows that the size descriptors are weakly linked to shape descriptors. In other words, although there is some connection between the molecular shape and its size, this connection is not strong. These results lead to the conclusion that while topological distance indices contain similar information about the molecular size, they contain less information about its shape. A comparison of performance between the 36 distance indices in structure – boiling point correlations for 74 alkanes of up to 9 carbon atoms showed that the most accurate QSPR models were obtained with VxDn (x = A, n = 3; x = A, n = 1,3) and ^kδ_λ (λ = 1, k = 1,2,3) topological indices and van der Waals descriptor of molecular size, i.e. volume (V^W) and surface (S^W).

Studies concerning the physical meaning of structural descriptors are therefore extremely useful. They allow the distillation of the informational content of such descriptors, preventing their misuse in QSPR studies.