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886 15. Information Processing – the Mission of Chemistry c) is independent of external conditions; d) is the least stable structure appearing in equilibrium conditions. 3. A molecular library composed of the associates of the molecules A and B represents: a) a mixture of the complexes AB; b) a mixture of all possible complexes of the A and B species; c) the complete physicochemical characterization of A and B; d) a mixture of all A n B n . 4. The self-organization of molecules is the spontaneous formation of: a) molecular complexes only in equilibrium conditions; b) a structure with minimum entropy; c) a structure with maximum entropy; d) complexes ofmolecules with synthons. 5. In the iterative solution of the logistic equation x n+1 =Kx n (1 −x n ): a) there is a fixed point at any K; b) at any attempt to increase of K we obtain a bifur- cation; c) some values of K lead to chaotic behaviour; d) at no value of K do we have extinction of the population. 6. In the Brusselator without diffusion the stable focus means: a) monotonic decreasing of the fluctuations x and y; b) dumped oscillations of the fluc- tuations x and y; c) non-vanishing oscillations of the fluctuations x and y; d) a limit circle. 7. In the thermodynamic equilibrium of an isolated system: a) the entropy increases; b) we may have a non-zero gradient of temperature; c) we may have a non-zero gradient of concentration; d) no dissipative structures are possible. 8. The bifurcation point for the number of solutions of x 2 −px +2 =0 corresponds to: a)p =2 √ 2; b) p =1; c) p =−1; d) p = √ 2. 9. An event has only four possible outputs with a priori probabilities p 1 =p 2 =p 3 =p 4 = 1 4 Reliable information comes that in fact the probabilities are different: p 1 = 1 2 , p 2 = 1 4 p 3 = 1 8 , p 4 = 1 8 The information had I 1 bits and I 1 is equal to: a) 1 bit; b) 05bit;c)2bits;d)0.25bit. 10. The situation corresponds to Question 9, but a second piece of reliable information coming says that the situation has changed once more and now: p 1 = 1 2 , p 2 =0p 3 = 0, p 4 = 1 2 The second piece of information had I 2 bits. We pay for information in proportion to its quantity. Therefore, for the second piece of information we have to pay: a) the same as for the first piece of information; b) twice as much as for the first piece of information; c) half of the prize for the first piece of information; d) three times more than for the first piece of information. Answers 1b, 2a, 3b, 4d, 5c, 6b, 7d, 8a, 9d, 10d APPENDICES This page intentionally left blank A. A REMINDER: MATRICES AND DETERMINANTS 1 MATRICES Definition A n × m matrix A represents a rectangular table of numbers 1 A ij standing like soldiers in n perfect rows and m columns (index i tells us in which row, and index j tells in which column the number A ij is located) A = ⎛ ⎜ ⎜ ⎝ A 11 A 12 A 1m A 21 A 22 A 2m A n1 A n2 A nm ⎞ ⎟ ⎟ ⎠ Such a notation allows us to operate whole matrices (like troops), instead of specifying what happens to each number (“soldier”) separately. If matrices were not invented, the equations would be very long and clumsy, instead they are short and clear. Addition Two matrices A and B may be added if their dimensions n and m match. The result is matrix C =A +B (of the same dimensions as A and B), where each element of C is the sum of the corresponding elements of A and B: C ij =A ij +B ij e.g., 1 −1 −34 + 21 −23 = 30 −57 Multiplying by a number A matrix may be multiplied by a number by multiplying every element of the matrix by this number: cA =B with B ij =cA ij For example, 2 1 −1 3 −2 = 2 −2 6 −4 1 If instead of numbers a matrix contained functions, everything below would remain valid (at partic- ular values of the variables instead of functions we would have their values). 889 890 A. A REMINDER: MATRICES AND DETERMINANTS Matrix product The product of two matrices A and B is matrix C denoted by C =AB, its elements are calculated using elements of A and B: C ij = N k=1 A ik B kj where the number of columns (N) of matrix A has to be equal to the number of rows of matrix B. The resulting matrix C has the number of rows equal to the number of rows of A and the number of columns equal to the number of columns of B. Let us see how it works in an example. The product AB =C: ⎛ ⎝ A 11 A 12 A 13 A 14 A 21 A 22 A 23 A 24 A 31 A 32 A 33 A 34 ⎞ ⎠ ⎛ ⎜ ⎜ ⎝ B 11 B 12 B 13 B 14 B 15 B 16 B 17 B 21 B 22 B 23 B 24 B 25 B 26 B 27 B 31 B 32 B 33 B 34 B 35 B 36 B 37 B 41 B 42 B 43 B 44 B 45 B 46 B 47 ⎞ ⎟ ⎟ ⎠ = ⎛ ⎝ C 11 C 12 C 13 C 14 C 15 C 16 C 17 C 21 C 22 C 23 C 24 C 25 C 26 C 27 C 31 C 32 C 33 C 34 C 35 C 36 C 37 ⎞ ⎠ e.g., C 23 is simply the dot product of two vectors or in matrix notation C 23 = A 21 A 22 A 23 A 24 · ⎛ ⎜ ⎜ ⎝ B 13 B 23 B 33 B 43 ⎞ ⎟ ⎟ ⎠ = A 21 B 13 +A 22 B 23 +A 23 B 33 +A 24 B 43 Some remarks: • The result of matrix multiplication depends in general on whether we have to multiply AB or BA, i.e. AB =BA. 2 • Matrix multiplication satisfies the relation (easy to check): A(BC) =(AB)C,i.e. the parentheses do not count and we can simply write: ABC. • Often we will have multiplication of a square matrix A by a matrix B composed of one column. Then, using the rule of matrix multiplication, we obtain matrix C in the form of a single column (with the number of elements identical to the dimension of A): ⎛ ⎜ ⎜ ⎝ A 11 A 12 A 1m A 21 A 22 A 2m A m1 A m2 A mm ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ B 1 B 2 B m ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ C 1 C 2 C m ⎞ ⎟ ⎟ ⎠ 2 Although it may happen, that AB =BA. 1Matrices 891 Transposed matrix For a given matrix A we may define the transposed matrix A T defined as (A T ) ij = A ji For example, if A = 12 −23 then A T = 1 −2 23 If matrix A = BC,thenA T = C T B T , i.e. the order of multiplication is reversed. Indeed, (C T B T ) ij = k (C T ) ik (B T ) kj = k C ki B jk = k B jk C ki = (BC) ji = (A T ) ij . Inverse matrix For some square matrices A (which will be called non-singular) we can define what is called the inverse matrix denoted as A −1 , which has the following prop- erty: AA −1 =A −1 A =1,where1 stands for the unit matrix: 1 = ⎛ ⎜ ⎜ ⎝ 10 0 01 0 00 1 ⎞ ⎟ ⎟ ⎠ For example, for matrix A = 20 03 we can find A −1 = 1 2 0 0 1 3 For square matrices A1 =1A =A. If we cannot find A −1 (because it does not exist), A is called a singular matrix. singular matrix For example, matrix A = 11 11 is singular. The inverse matrix for A = BC is A −1 =C −1 B −1 .Indeed,AA −1 =BCC −1 B −1 =B1B −1 =BB −1 =1. Adjoint, Hermitian, symmetric matrices If matrix A is transposed and in addition all its elements are changed to their com- plex conjugate, we obtain the adjoint matrix denoted as A † = (A T ) ∗ = (A ∗ ) T If, for a square matrix, we have A † =A, A is called Hermitian.IfA is real, then, of course, A † =A T . If, in addition, for a real square matrix A T =A,thenA is called symmetric. Examples: A = 1 +i 3 −2i 2 +i 3 −i ; A T = 1 +i 2 +i 3 −2i 3 −i ; A † = 1 −i 2 −i 3 +2i 3 +i Matrix A = 1 −i i −2 is an example of a Hermitian matrix, because A † = A. Matrix A = 1 −5 −5 −2 is a symmetric matrix. 892 A. A REMINDER: MATRICES AND DETERMINANTS Unitary and orthogonal matrices If for a square matrix A we have A † = A −1 , A is called a unitary matrix. If B is Hermitian, the matrix exp(iB) is unitary, where we define exp(iB) by using the Taylor expansion: exp(iB) =1+iB+ 1 2! (iB) 2 +···.Indeed,[exp(iB)] † =1−iB T + 1 2! (−iB T ) 2 +···=1 − iB + 1 2! (−iB) 2 +···=exp(−iB),whileexp(iB) exp(−iB) =1. If A is a real unitary matrix A † = A T , it is called orthogonal with the property A T =A −1 . For example, if A = cosθ sinθ −sinθ cosθ then A T = cosθ −sinθ sinθ cosθ =A −1 Indeed, AA T = cosθ sinθ −sinθ cosθ cosθ −sinθ sinθ cosθ = 10 01 2 DETERMINANTS Definition For any square matrix A ={A ij } we may calculate a number called its determinant and denoted by detA or |A|. The determinant is calculated by using the Laplace expansion detA = N i (−1) i+j A ij ¯ A ij = N j (−1) i+j A ij ¯ A ij where (N is the dimension of the matrix) the result does not depend on which column j has been chosen in the first expression or which row i in the second expression. The symbol ¯ A ij stands for the determinant of the matrix, which is ob- tained from A by removing the i-th row and the j-th column. Thus we have defined a determinant (of dimension N) by saying that it is a certain linear combination of determinants (of dimension N − 1). It is then sufficient to say what we mean by a determinant that contains only one number c (i.e. has only one row and one column), this is simply detc ≡c. For example, for matrix A = ⎛ ⎝ 10−1 22 4 3 −2 −3 ⎞ ⎠ its determinant is detA = 10−1 22 4 3 −2 −3 = (−1) 1+1 ×1 × 24 −2 −3 +(−1) 1+2 ×0 × 24 3 −3 +(−1) 1+3 ×(−1) × 22 3 −2 = 24 −2 −3 − 22 3 −2 2 Determinants 893 = 2 ×(−3) −4 ×(−2) − 2 ×(−2) −2 ×3 = 2 +10 =12 In particular, ab cd =ad −bc. By repeating the Laplace expansion again and again (i.e. expanding ¯ A ij ,etc.) we finally arrive at a linear combination of the products of the elements detA = P (−1) p ˆ P[A 11 A 22 ···A NN ] where the permutation operator ˆ P pertains to the second indices (shown in bold), and p is the parity of permutation ˆ P. Slater determinant In this book we will most often have to do with determinants of matrices whose elements are functions, not numbers. In particular what are called Slater determi- nants will be the most important. A Slater determinant for the N -electron system is built of functions called spinorbitals φ i (j), i = 1 2N, where the symbol j denotes the space and spin coordinates (x j y j z j σ j ) of electron j: ψ(1 2N)= φ 1 (1)φ 1 (2) φ 1 (N) φ 2 (1)φ 2 (2) φ 2 (N) φ N (1)φ N (2) φ N (N) After this is done the Laplace expansion gives ψ(1 2N)= P (−1) p ˆ P φ 1 (1)φ 2 (2)φ N (N) where the summation is over N! permutations of the N electrons, ˆ P stands for the permutation operator that acts on the arguments of the product of the spinorbitals [φ 1 (1)φ 2 (2)φ N (N)], p is the parity of permutation ˆ P (i.e. the number of the transpositions that change [φ 1 (1)φ 2 (2)φ N (N)]into ˆ P[φ 1 (1)φ 2 (2)φ N (N)]. All properties of determinants pertain also to Slater determinants. Some useful properties • detA T =det A. • From the Laplace expansion it follows that if one of the spinorbitals is composed of two functions φ i =ξ +ζ, the Slater determinant is the sum of the two Slater determinants, one with ξ instead of φ i , the second with ζ instead of φ i • If we add to a row (column) any linear combination of other rows (columns), the value of the determinant does not change. 894 A. A REMINDER: MATRICES AND DETERMINANTS • If a row (column) is a linear combination of other rows (columns), then detA = 0. In particular, if two rows (columns) are identical then detA =0. Conclusion: in a Slater determinant the spinorbitals have to be linearly independent, other- wise the Slater determinant is equal zero. • If in a matrix A we exchange two rows (columns), then det A changes sign. Con- clusion: the exchange of the coordinates of any two electrons leads to a change of sign of the Slater determinant (Pauli exclusion principle). • det(AB) =detA detB. • From the Laplace expansion it follows that multiplying the determinant by a number is equivalent to multiplication of an arbitrary row (column) by this num- ber. Therefore, det(cA) =c N detA,whereN is the matrix dimension. 3 • If matrix U is unitary then detU = exp(iφ),whereφ is a real number. This means that if U is an orthogonal matrix, det U =±1. 3 Note, that to multiply a matrix by a number we have to multiply every element of the matrix by this number. However, to multiply a determinant by a number means multiplication of one row (column) by this number. B. A FEW WORDS ON SPACES, VECTORS AND FUNCTIONS 1 VECTOR SPACE A vector space means a set V of elements xy, that form an Abelian group and can be “added” together 1 and “multiplied” by numbers z =αx +βy thus produc- ing z ∈V . The multiplication (α β are, in general, complex numbers) satisfies the usual rules (the group is Abelian, because x +y =y +x): 1 ·x =x α(βx) =(αβ)x α(x +y) =αx +αy (α +β)x =αx +βx Example 1. Integers. The elements xy are integers, the “addition” means simply the usual addition of integers, the numbers αβare also integers, “mul- tiplication” means just usual multiplication. Does the set of integers form a vector space? Let us see. The integers form a group (with the addition as the operation in the group). Checking all the above axioms, we can easily prove that they are satisfied by integers. Thus, the integers (with the operations defined above) form a vector space. Example 2. Integers with real multipliers. If, in the previous example, we admitted α β to be real, the multiplication of integers x y by real numbers would give real numbers (not necessarily integers). Therefore, in this case xydo not represent any vector space. Example 3. Vectors. Suppose xy are vectors, each represented by a N- element sequence of real numbers (they are called the vector “components”) x =(a 1 a 2 a N ), y =(b 1 b 2 b N ) etc. Their addition x+y is an operation that produces the vector z =(a 1 +b 1 a 2 +b 2 a N +b N ). The vectors form an Abelian group, because x +y =y +x, the unit (“neutral”) element is (000), the inverse element to (a 1 a 2 a N ) is equal to (−a 1 −a 2 −a N ).Thus, the vectors form a group. “Multiplication” of a vector by a real number α means α(a 1 a 2 a N ) =(αa 1 αa 2 αa N ). Please check that the four axioms above are satisfied. Conclusion: the vectors form a vector space. 1 See Appendix C, to form a group any pair of the elements can be “added” (operation in the group), the addition is associative, there exists a unit element and for each element an inverse exists. 895 . elements of A and B: C ij = N k=1 A ik B kj where the number of columns (N) of matrix A has to be equal to the number of rows of matrix B. The resulting matrix C has the number of rows equal. represents: a) a mixture of the complexes AB; b) a mixture of all possible complexes of the A and B species; c) the complete physicochemical characterization of A and B; d) a mixture of all A n B n . 4 second piece of information we have to pay: a) the same as for the first piece of information; b) twice as much as for the first piece of information; c) half of the prize for the first piece of information;
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