Properties of determinants

The determinant of a matrix A, Det(A), is useful in analyzing the uniqueness of a solution for a system of linear equations. Determinants arise in AR theory when computing conditions for critical CSTRs and DSRs. A number of properties of determinants are provided in the following text. Many of these properties are used in Chapters 6 and 7. 

Row operations on A do not affect the value of its determinant. A multiple of a row or column in A added to another row or column in A does not affect Det(A). 

A determinant is unaltered in value if all rows and columns are interchanged, e.g.  

the factor 2 has been removed from column 2. Conversely, when a determinant is multiplied by a constant, the constant can be absorbed into the determinant by multiplying the elements of one row (or column) by that constant. 

The value of a determinant is unaltered if a constant multiple of one row or column is added to or subtracted from another row or column, respectively. For example, if we subtract twice column 1 from column 2, we obtain  

A determinant can only have a value if the elements are numbers. 

Any matrix A and its transpose have the same determinant, meaning 

The determinant of a triangular matrix is the product of the entries on the diagonal, that is. 

If one determinant is obtained from another by interchanging any two rows (or columns), tire value of either is the negative of the value of the other. 

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The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

The expansion of this determinant is identical (1, 2,. .., N) given by (8.47). The properties of determinants are discussed in Appendix 1. The wave function in equation (8.51) is clearly antisymmetric because interchanging any pair of particles is equivalent to interchan-... 

Some general properties of determinants can be summarized as follows. 

Antisymmetry is a requirement of acceptable solutions to the Schrodinger equation. The fact that the determinant form satisfies this requirement follows from the fact that different electrons correspond to different rows in the determinant. Interchanging the coordinates of two electrons is, therefore, equivalent to interchanging two rows in the determinant which, according to the properties of determinants, multiplies the value of the determinant by -1. 

The fact that there are only two kinds of spin function (a and (1), leads to the conclusion that two electrons at most may occupy a given molecular orbital. Were a third electron to occupy the orbital, two different rows in the determinant would be the same which, according to the properties of determinants, would cause it to vanish (the value of the determinant would be zero). Thus, the notion that electrons are paired is really an artifact of the Hartree-Fock approximation. 

It follows from the properties of determinants that, if the entire determinant is to have the value zero, each block factor separately must equal zero. Thus the 10 x 10 determinantal equation has been reduced to two 2x2 and two 3x3 secular equations. For example, the energies of the two MOs of Au symmetry are given by the simple secular equation... 

Another well-known property of determinants is that they vanish if they have two identical rows. This means that it is not possible to construct a non-vanishing antisymmetrised product in which two electrons in the same orbital have the same spin. Thus the rule that not more than two electrons must be assigned to any one space orbital follows as a direct consequence of the antisymmetry principle for product wave functions it had to be introduced as an extra postulate. 

The antisymmetry principle is also of great importance in understanding the dualism between localised and delocalised descriptions of electronic structure. We shall see that these are just different ways of building up the same total determinantal wave functions.1 This can be developed mathematically from general properties of determinants, but a clearer picture can be formed if we make a detailed study of the antisymmetric wave function for some highly simplified model systems. 

Due to the properties of determinants, a Slater determinantal wavefunction ° automatically fulfils the Pauli principle and takes care of the antisymmetric character of fermions. If written explicitly in terms of the single-particle orbitals,... 

It is clear from the above discussion that the difference between a PPD and its corresponding determinant solely lies in the coefficients of the permutation P. Unfortunately, this makes PPDs unable to share many of the nice properties of determinants. For instance, the basic multiplicative law valid for determinants... 

The most impressive property of determination is the extraordinary stability of its consequences. The process takes only a few hours to... 

The variety of chemical and physical properties of determined trace components required rapid development of both gas and liquid chromatographic instrumentation. Many laboratories worked on the development and optimization of the structure and composition of stationary and mobile phases according to the properties, structure, and size of separated molecules. Studies were devoted to correlation of their physicochemical characteristics with chromatographic parameters. For typical applications, numerous commercial products and new instrument versions were developed. 

The properties of determinants are more easily grasped by considering determinants of the second and third orders, when n=2 and =3. (A single symbol, e.g. ai, may be formally regarded as a determinant of the first order.)... 

Note that the commutation rules (3.137—3.139) and the symmetric operators (3.142,3.149) have been derived from properties of determinants. We have not assumed that the orbitals p), v) are orthogonal. In evaluating matrix elements care must be taken to keep track of the scalar products of orbitals that are not orthogonal, such as bound orbitals and plane waves. The iV-electron target configurations are conveniently normalised by (3.123). The normalisation of the continuum orbitals is discussed in chapter 6. 

There is another property of determinants which is important to consider determinants remain unchanged when the respective elements are subject to some specific operations a unitary transformation) as can easily be illustrated in Problem 5.2. 

That R. — r for irratrix ip cair be deiirorrstrated if orre critploys the following properties of determinants, w hich we summarize without proof as follows ... 

However, beeaiLse of Eq. (B.47) the elements of the nth column are identical to the ones forming column n+1 regardless of k such that all (n+1) x (n+1) subdeterminants of ip vanish on account of statement 4 of our list of elementary properties of determinants. This, in turn, proves that the rank of matrix ip is equal to the rank of matrix p. ... 

Use the properties of determinants to introduce as many zeros as possible to the right (or left) of the leading diagonal... 

Properties of determinants, and their use to simplify the expansion of determinants of high order. 

Using the properties of determinants, we see that exchanging rows two and three results in a change of sign of the determinant such a change corresponds to the vector product c x b and is consistent with equation (5.15) and b x c = —cxb. 

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