## 3 Electronic Communications  
In OCT of the chemical bond [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 62], one explores the entropy/information descriptors of molecular information channels, e.g., the electron probability networks in the orthonormal Atomic Orbital (AO) resolution \(|\mathbf{x}\rangle=\{|\chi_{j}\rangle,\quad j=1,\,2,\,\dots,\,t\}\),  
\[\langle\chi_{i}|\chi_{j}\rangle=\int\!\langle\chi_{i}|\mathbf{r}\rangle\;\langle \mathbf{r}|\chi_{j}\rangle\;d\mathbf{r}=\int\!\chi_{i}(\mathbf{r})^{*}\,\chi_{j}(\mathbf{r})\;d \mathbf{r}=\delta_{i,j}. \tag{21}\]Such electron communications are reflected by the conditional probabilities of observing in the molecular bond system the monitored \"output\" orbitals, given the specified \"input\" orbitals. In this standard LCAO MO approach, the optimum MO \(|\varphi\rangle=\{|\varphi_{w}\rangle,\;\;w=1,\,2,\,\ldots,\,t\}\) are represented as Linear Combinations of the AO basis functions:  
\[\varphi(\mathbf{r})=\{\varphi_{w}(\mathbf{r})=\Sigma_{j}\;\chi_{j}(\mathbf{r})\;C_{j,w} \}=\mathbf{\chi}(\mathbf{r})\;\mathbf{C},\;\;\;\;\;\;\;\;\;\;\;\mathbf{\chi}(\mathbf{r})=\langle \mathbf{r}|\mathbf{\chi}\rangle, \tag{22}\]  
where the square matrix of the expansion coefficients, \(\mathbf{C}=\langle\mathbf{\chi}|\varphi\rangle=\{C_{i,w}\}\), satisfies the matrix orthonormality relations:  
\[\mathbf{C}\mathbf{C}^{\dagger}=\{\delta_{i,j}\}=\mathbf{C}^{\dagger}\mathbf{C} =\{\delta_{w,w^{\prime}}\}\equiv\mathbf{I}. \tag{23}\]  
The AO-resolved communication theory explores the entropy/information descriptors of the electronic-signal propagation network between \"input\" and \"output\" AO states. These scattering connections in the given electron configuration of Equation (20) are determined by its occupied MO, \(\psi(\mathbf{\Psi})\). The representative probability of observing in \(\mathbf{\Psi}\) the specified output AO \(\chi_{j}(\mathbf{r})=\langle\mathbf{r}|\chi_{j}\rangle=m_{j}(\mathbf{r})\;\exp[i\phi_{j}( \mathbf{r})]\), given the input AO \(\chi_{i}(\mathbf{r})=\langle\mathbf{r}|\chi_{i}\rangle=m_{i}(\mathbf{r})\;\exp[i\phi_{i}( \mathbf{r})]\),  
\[P(\chi_{j}|\chi_{i})=P(\chi_{i},\chi_{j})/P(\chi_{i})\equiv P(j|i),\;\;\;\;\; \;\;\;\Sigma_{j}\;P(j|i)=1, \tag{24}\]  
is generated by the squared modulus of the associated conditional amplitude \(A(\chi_{j}|\chi_{i})\equiv A(j|i)\):  
\[\{P(j|i)\equiv|A(j|i)|^{2}\}. \tag{25}\]  
Here, \(P(\chi_{i})\equiv P(i)\) denotes the probability of detecting AO state \(|\chi_{i}\rangle\equiv|i\rangle\) in the chemical bond system of the electron configuration in question, while \(P(\chi_{i},\chi_{j})\equiv P(i,j)\) stands for the _joint_-probability of these two AO events, of simultaneously observing the two AO states \(|i\rangle\) and \(|j\rangle\) in \(\mathbf{\Psi}\). These probabilities satisfy the relevant normalizations:  
\[\Sigma_{i}\;[\Sigma_{j}\;P(i,j)]=\Sigma_{i}\;P(i)=1. \tag{26}\]  
In accordance with the Superposition Principle (SP) of QM [93], the scattering amplitude \(A(\zeta|\theta)\) between states \(\theta(\mathbf{r})\equiv\langle\mathbf{r}|\theta\rangle=m_{\theta}(\mathbf{r})\;\exp[i \phi_{\theta}(\mathbf{r})]\) and \(\zeta(\mathbf{r})\equiv\langle\mathbf{r}|\zeta\rangle=m_{\zeta}(\mathbf{r})\;\exp[i\phi_{ \zeta}(\mathbf{r})]\) is defined by the mutual projection of the two state-vectors involved:  
\[ \begin{array}{l}A(\zeta|\theta)=\langle\theta|\zeta\rangle\equiv\int\! \theta(\mathbf{r})^{*}\;\zeta(\mathbf{r})\;dr=\int\!m_{\theta}(\mathbf{r})\;m_{\zeta}(\mathbf{ r})\;\exp[i[\phi_{\zeta}(\mathbf{r})-\phi_{\theta}(\mathbf{r})]]\;d\mathbf{r}\\ \equiv\int\!M(\zeta|\theta)\;\exp[i\phi(\zeta|\theta)]\;d\mathbf{r}.\end{array}