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Ideas of Quantum Chemistry P94

Ideas of Quantum Chemistry P94 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 896 B. A FEW WORDS ON SPACES VECTORS AND FUNCTIONS Note that if only the positive vector components were allowed they would not form an Abelian group no neutral element and on top of this their addition which might mean a subtraction of components because a ft could be negative could produce vectors with non-positive components. Thus vectors with all positive components do not form a vector space. Example 4. Functions. This example is important in the context of this book. This time the vectors have real components.2 Their addition means the addition of two functions f x fi x f2 x . The multiplication means multiplication by a real number. The unit neutral function means f 0 the inverse function to f is -f x . Therefore the functions form an Abelian group. A few seconds are needed to show that the four axioms above are satisfied. Such functions form a vector space. Linear independence. A set of vectors is called a set of linearly independent vectors if no vector of the set can be expressed as a linear combination of the other vectors of the set. The number of linearly independent vectors in a vector space is called the dimension of the space. Basis means a set of n linearly independent vectors in n-dimensional space. 2 EUCLIDEAN SPACE A vector space with multiplying real numbers a ft represents the Euclidean space if for any two vectors x y of the space we assign a real number called an inner or scalar product x y with the following properties x y y x ax y a x y xi x2 y xi y x2 y x x 0 only if x 0. Inner product and distance. The concept of the inner product is used to introduce the length of the vector x defined as x Vx x and the distance between two vectors x and y as a non-negative number x - y V x - y x - y . The distance satisfies some conditions which we treat as obvious from everyday experience the distance from Paris to Paris has to equal zero just insert x y the distance from Paris to Rome has to be the same as from Rome to Paris just exchange x y the .