Homogeneous simultaneous equations

What we fomierly called the nonhomogeneous vector (Chapter 2) is zero in the pair of simultaneous nomial equations Eq. set (6-38). When this vector vanishes, the pair is homogeneous. Let us try to construct a simple set of linearly independent homogeneous simultaneous equations. 

These equations cannot be true for any solution set other than 0, 0. The determinant of the coefficients is not zero 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

Linearly dependent sets of homogeneous simultaneous equations, for example. 

To select one from among the infinite number of solution sets, we must have an additional independent nonhomogeneous equation. If the additional equation is 

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Note that the integrals ny are evaluated with the known initial set of unperturbed eigenfunctions, in contrast to the integrals in equation (9.63), which require the unknown functions 0. For a given eigenvalue E, the expression (9.64) is a set of gn linear homogeneous simultaneous equations, one for each value of / (j3 = 1, 2,..., gn)... 

Linear homogeneous simultaneous equations have a nontrivial solution only when a certain dependence condition is met. 

This is the secular equation in matrix form using cartesian displacement coordinates. It is a series of linear homogeneous simultaneous equations [similar to Eq. (14.8)] from which A and A can be determined. 

In various methods in quantum chemistry, orbital functions are represented as linear combinations of basis functions. A set of linear homogeneous simultaneous equations arises that is to be solved for the coefficients in the linear combinations. The determinant condition is... 

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