Orbital overlap

<h2 id="overlap-matrix">Overlap matrix</h2>
<p>The overlap matrix is a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a>, used in <a href="/facts/Quantum_chemistry/H9AXV0lx">quantum chemistry</a> to describe the inter-relationship of a set of <a href="/facts/Basis_vector/89IPoN6c">basis vectors</a> of a <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum</a> system, such as an atomic orbital <a href="/facts/Basis_set_(chemistry)/rQCPt3yl">basis set</a> used in molecular electronic structure calculations. In particular, if the vectors are <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an <a href="/facts/Orthonormal/JfgL1nAh">orthonormal</a> set, the overlap matrix will be the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. The overlap matrix is always <i>n</i>×<i>n</i>, where <i>n</i> is the number of basis functions used. It is a kind of <a href="/facts/Gramian_matrix/SGb2VGTU">Gramian matrix</a>.
</p><p>In general, each overlap matrix element is defined as an overlap integral:
</p>

S
          
          
            j
            k
          
        
        =
        
          ⟨
          
            
              b
              
                j
              
            
            
              |
            
            
              b
              
                k
              
            
          
          ⟩
        
        =
        ∫
        
          Ψ
          
            j
          
          
            ∗
          
        
        
          Ψ
          
            k
          
        
        
        d
        τ
      
    
    {\displaystyle \mathbf {S} _{jk}=\left\langle b_{j}|b_{k}\right\rangle =\int \Psi _{j}^{*}\Psi _{k}\,d\tau }

<p>where
</p>

|
          
            b
            
              j
            
          
          ⟩
        
      
    
    {\displaystyle \left|b_{j}\right\rangle }
  
 is the <i>j</i>-th basis <a href="/facts/Bra-ket_notation/NDlKsaRh">ket</a> (<a href="/facts/Vector_(geometric)/bCLrdksy">vector</a>), and

Ψ
          
            j
          
        
      
    
    {\displaystyle \Psi _{j}}
  
 is the <i>j</i>-th <a href="/facts/Wavefunction/KgM5ykjY">wavefunction</a>, defined as :
  
    
      
        
          Ψ
          
            j
          
        
        (
        x
        )
        =
        
          ⟨
          
            x
            
              |
            
            
              b
              
                j
              
            
          
          ⟩
        
      
    
    {\displaystyle \Psi _{j}(x)=\left\langle x|b_{j}\right\rangle }
  
.
<p>In particular, if the set is normalized (though not necessarily orthogonal) then the diagonal elements will be identically 1 and the magnitude of the <a href="/facts/Off-diagonal_element/39z9sIRb">off-diagonal elements</a> less than or equal to one with equality if and only if there is linear dependence in the basis set as per the <a href="/facts/Cauchy%E2%80%93Schwarz_inequality/poMzQUIY">Cauchy–Schwarz inequality</a>.  Moreover, the matrix is always <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive definite</a>; that is to say, the eigenvalues are all strictly positive.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Roothaan_equations/RLDmxFfC">Roothaan equations</a></li>
<li><a href="/facts/Hartree%E2%80%93Fock_method/U9mlRWVR">Hartree–Fock method</a></li>
<li><a href="/facts/Pi_bond/3Q6LAGYK">Pi bond</a></li>
<li><a href="/facts/Sigma_bond/CSyEe5CY">Sigma bond</a></li></ul>

<p><i>Quantum Chemistry: Fifth Edition</i>, Ira N. Levine, 2000
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Anslyn, Eric V./Dougherty, Dennis A. (2006).  Modern Physical Organic Chemistry.  University Science Books. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Pauling, Linus.  (1960).  The Nature Of The Chemical Bond.  Cornell University Press. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Orbital overlap open-in-new

Orbital overlap