1 The QQSPR fundamental equation involving a QMP and a collection of different properties

The quantum QSPR (QQSPR) fundamental equation can be deduced from the collective distances [1–3], which can be defined by taking into account the geometrical structure of any QMP [4], which is nothing else that a geometrical picture associated to the DF tag set, forming a quantum object set [5].

The general QQSPR form of the fundamental equation can be written as [4]:

$$\begin{aligned} \forall I=1,M_K :\pi _I \approx \sum _P {\omega _P} A_{{ PI}} , \end{aligned}$$
(1)

where \(M_K\) is the number of molecules involved belonging to a QMP and possessing known values of some property \(\pi \), which can be supposed that are collected in the set \(\Pi =\left\{ {\pi _I \left| {I=1,M_K} \right. } \right\} \). On the QQSPR equation right side the set of coefficients \(\hbox {O}=\left\{ {\omega _P \left| {P=0,2,3,...} \right. } \right\} \) might be determined to an arbitrary order. Finally, the set of matrix elements \(\mathbf{A}=\left\{ {A_{{ PI}} \left| {P=0,2,3,...\wedge I=1,M_K}\right. }\right\} \) correspond to the \({{ P}\mathrm{th}}\) order element of the collective QMP condensed index associated to the \({{ I}\mathrm{th}}\) molecule [3].

The collective QMP condensed indices can be obtained from the expression of the \({{ P}\mathrm{th}}\) order QMP collective moment, related to QMP collective distances [6, 7], which can be basically constructed with the origin shifted DF set, see for example reference [8], expression:

$$\begin{aligned} \forall P{:}\,\mu _P =M^{-1}\sum _L {\left( {\rho _L -\rho _C } \right) ^{P}} \end{aligned}$$
(2)

where \(\rho _C\), the QMP centroid, is simply defined as the arithmetic mean [8]:

$$\begin{aligned} \rho _C =M^{-1}\sum _L {\rho _L } . \end{aligned}$$

Note that the total number of QMP DF vertices: M, might be different from those which possess a known property attached, \(M_K\). In fact, the total number of QMP vertices \(M=M_K +M_U\), can be split into those \(M_K\) with known property attached and those \(M_U\) with unknown property values.

Condensing expression (2) corresponds to obtain the integral of the DF moment [2, 3]:

$$\begin{aligned} \forall P{:}\left\langle {\mu _P } \right\rangle =M^{-1}\sum _L {\left\langle {\left( {\rho _L -\rho _C } \right) ^{P}} \right\rangle } \end{aligned}$$

and then it can be written:

$$\begin{aligned} \forall P,L=1,M:A_{{ PL}} =\left\langle {\left( {\rho _L -\rho _C } \right) ^{P}} \right\rangle . \end{aligned}$$

It can be also easily shown [4] that solving Eq. (1) to obtain the coefficients in the set: \(\hbox {O}\), becomes equivalent to describe some QQSPR operator \(\Omega \), which generally speaking can fulfill:

$$\begin{aligned} \forall L=1,M:\left\langle {\Omega \rho _L } \right\rangle =\pi _L , \end{aligned}$$

supposedly known the DF set \(\hbox {P}=\left\{ {\rho _L \left| {L=1,M} \right. } \right\} \), associated to the QMP DF vertices. This quantum mechanical expectation value expression above will also hold, whenever the QQSPR operator can be written as a linear combination of a set of Hermitian operators:

$$\begin{aligned} \Omega \approx \sum _P {\omega _P \Omega _P}; \end{aligned}$$

where, besides the elements of the coefficient set \(\hbox {O}\), it must be taken into account that:

$$\begin{aligned} \Omega _0 =I\wedge \Omega _1 =0 \end{aligned}$$

and

$$\begin{aligned} \forall P=0,2,3...\wedge L=1,M{:}\left\langle {\Omega _P \rho _L } \right\rangle =A_{{ PL}} . \end{aligned}$$

In order to solve Eq. (1) one can set up an equivalent matrix equation, which can be written with the same form, but using only the \(M_K\) elements of the condensed QMP collective indices, associated to known properties:

$$\begin{aligned} \mathbf{A}\left| \mathbf{w} \right\rangle =\left| {\varvec{\uppi }} \right\rangle \end{aligned}$$
(3)

where it has been used the formerly defined matrix \(\mathbf{A}\), the column vectors collecting the QQSPR operator coefficients: \(\left| \mathbf{w} \right\rangle =\left\{ {\omega _P } \right\} \) and the set known property values: \(\left| {\varvec{\uppi }} \right\rangle =\left\{ {\pi _I \left| {I=1,M_K } \right. } \right\} \).

Equation (3) can be solved via a least squares procedure, which provides the intermediate equation:

$$\begin{aligned} \mathbf{A}^{T}\mathbf{A}\left| \mathbf{w} \right\rangle =\mathbf{A}^{T}\left| {\varvec{\uppi }} \right\rangle , \end{aligned}$$
(4)

obtained by multiplying Eq. (1) on the left by the transpose matrix: \(\mathbf{A}^{T}\). The well-known optimal result for the coefficients of the QQSPR operator might be easily obtained via:

$$\begin{aligned} \left| \mathbf{w} \right\rangle =\left( {\mathbf{A}^{T}\mathbf{A}} \right) ^{-1}\mathbf{A}^{T}\left| {\varvec{\uppi }} \right\rangle . \end{aligned}$$
(5)

2 General QQSPR equation

Equation (3) and the least squares optimal solution (5) can be easily generalized when, instead of the property column vector \(\left| {\varvec{\uppi }} \right\rangle \), a set of properties is known in form of a matrix, whose columns are vectors corresponding to the set of Q different properties, that is:

$$\begin{aligned} {\varvec{\Pi }} =\left\{ {\left| {{\varvec{\uppi }}_R } \right\rangle \left| {R=1,Q} \right. } \right\} \end{aligned}$$

Therefore, the coefficient vector \(\left| \mathbf{w} \right\rangle \) associated to the QQSPR operator might be transformed into a matrix, whose columns correspond to a QQSPR operator adequate to every property considered:

$$\begin{aligned} \mathbf{W}=\left\{ {\left| {\mathbf{w}_R } \right\rangle \left| {R=1,Q} \right. } \right\} . \end{aligned}$$

Within this general formalism, Eq. (3) becomes:

$$\begin{aligned} \mathbf{AW}={\varvec{\Pi }} \end{aligned}$$
(6)

and the corresponding optimal solution in the least squares sense will be converted into:

$$\begin{aligned} \mathbf{W}=\left( {\mathbf{A}^{T}\mathbf{A}} \right) ^{-1}\mathbf{A}^{T}{\varvec{\Pi }}. \end{aligned}$$
(7)

3 Estimating unknown property values and the nature of the optimal least squares solution of the QQSPR equation

Equation (7) can be used in Eq. (6) in order to obtain unknown properties for some of QMP DF vertices, which have not been attached to any known property, and then it can be written:

$$\begin{aligned} \mathbf{A}_U \mathbf{W}\approx {\varvec{\Pi }}_U \rightarrow \mathbf{A}_U \left( {\mathbf{A}_K^T \mathbf{A}_K } \right) ^{-1}\mathbf{A}_K^T {\varvec{\Pi }}_K \approx {\varvec{\Pi }}_U . \end{aligned}$$
(8)

This result corresponds to an expression, which indicates that the unknown estimated property values just become the result of transforming, via a matrix operator, the known ones. To see how this statement can hold, it is sufficient to construct the matrix:

$$\begin{aligned} \mathbf{T}_{{ UK}} =\mathbf{A}_U \left( {\mathbf{A}_K^T \mathbf{A}_K } \right) ^{-1}\mathbf{A}_K^T ; \end{aligned}$$
(9)

then, it is straightforward to write:

$$\begin{aligned} \mathbf{T}_{{ UK}} {\varvec{\Pi }}_K ={\varvec{\Pi }}_U ; \end{aligned}$$

therefore, the matrix (9) corresponds to the matrix expression of some operator, which is able to transform vectors belonging to the known property values set into vectors which belong to the unknown property values set.

4 Accuracy of the least squares solution of the QQSPR fundamental equation

When it is used the same matrix \(\mathbf{A}_K\) into Eq. (6) instead of \(\mathbf{A}_U\), then expression (8) can be rewritten as:

$$\begin{aligned} \mathbf{A}_K \mathbf{W}\approx {\varvec{\Pi }}_K \rightarrow \mathbf{A}_K \left( {\mathbf{A}_K^T \mathbf{A}_K } \right) ^{-1}\mathbf{A}_K^T {\varvec{\Pi }}_K \approx {\varvec{\Pi }}_K^{calc} \end{aligned}$$

and thus, whenever it can be written:

$$\begin{aligned} {\varvec{\Pi }}_K \approx {\varvec{\Pi }}_K^{calc} , \end{aligned}$$
(10)

also it can be obtained:

$$\begin{aligned} \mathbf{A}_K \left( {\mathbf{A}_K^T \mathbf{A}_K } \right) ^{-1}\mathbf{A}_K^T {\varvec{\Pi }}_K -{\varvec{\Pi }}_K^{calc} \approx \mathbf{0}\rightarrow \left( {\mathbf{A}_K \left( {\mathbf{A}_K^T \mathbf{A}_K } \right) ^{-1}\mathbf{A}_K^T -\mathbf{I}} \right) {\varvec{\Pi }}_K \approx \mathbf{0}. \end{aligned}$$

Taking into account that the original property matrix is not null by construction, so: \({\varvec{\Pi }}_K \ne \mathbf{0}\), then it can be also considered that necessarily:

$$\begin{aligned} \mathbf{A}_K \left( {\mathbf{A}_K^T \mathbf{A}_K} \right) ^{-1}\mathbf{A}_K^T -\mathbf{I}\approx \mathbf{0}\rightarrow \mathbf{A}_K \left( {\mathbf{A}_K^T \mathbf{A}_K } \right) ^{-1}\mathbf{A}_K^T \approx \mathbf{I}\rightarrow \mathbf{T}_{{ KK}} \approx \mathbf{I}. \end{aligned}$$

Then it can be easily deduced that the accuracy of the calculation, represented by the Eq. (10), and which has to be associated to the endomorphism:

$$\begin{aligned} \mathbf{T}_{{ KK}} {\varvec{\Pi }}_K ={\varvec{\Pi }}_K^{calc}, \end{aligned}$$

has to be obviously associated to the nearness of the endomorphism \(\mathbf{T}_{{ KK}}\) to the unit matrix. This will be exactly fulfilled when the matrix \(\mathbf{A}_K\) becomes non-singular, and thus it might be written:

$$\begin{aligned} \left( {\mathbf{A}_K^T \mathbf{A}_K } \right) ^{-1}=\mathbf{A}_K^{-1} \left( {\mathbf{A}_K^T } \right) ^{-1}\rightarrow \mathbf{T}_{{ KK}} =\mathbf{I}. \end{aligned}$$
(11)

In this case, because Eq. (11) holds, then Eq. (7) can be written in a simplified form as:

$$\begin{aligned} \mathbf{W}=\mathbf{A}_K^{-1} {\varvec{\Pi }}_K , \end{aligned}$$

therefore Eq. (8) can be also simplified in this circumstance to:

$$\begin{aligned} \mathbf{A}_U \mathbf{W}\approx {\varvec{\Pi }}_U \rightarrow \mathbf{A}_U \mathbf{A}_K^{-1} {\varvec{\Pi }}_K \approx {\varvec{\Pi }}_U \end{aligned}$$

and the transformation operator (9) now could be written simply as:

$$\begin{aligned} \mathbf{T}_{{ UK}} =\mathbf{A}_U \mathbf{A}_K^{-1} . \end{aligned}$$

5 Conclusions

Constructing a QMP, where some DF vertices might be also associated to a set of property values, an approximate set of operators can be obtained. This can be done just constructing the QQSPR fundamental equation and solving it via a general least squares technique. The QQSPR operators can be employed afterwards to evaluate the properties of the remnant QMP DF vertices possessing unknown property values.

In setting up and solving in this way the problem, one has several points to note, which can be found among the characteristics of QQSPR fundamental equation results [4]:

  1. 1.

    Nothing opposes to the fact that a general QQSPR equation might be built using known molecular property values attached to some QMP DF vertices. Its solution permitting to obtain a set of Hermitian operators, which can be further used to estimate values of molecular properties, unknown in other molecular structures forming the QMP.

  2. 2.

    This previous fact corresponds to establishing a causal relationship between QMP structures and aromaticity descriptors [9].

  3. 3.

    The basis of this relationship is based into quantum mechanical theoretical grounds.

Whenever Eq. (6) can be solved yielding results of the recomputed properties within reasonable error limits, and the expression (7) defining the QQSPR operators is reachable, then one can be confident that there has been obtained a global theoretical expression, which permits to estimate unknown values of molecular properties, constructed within the quantum mechanical limits of the theory.