| Home / lib / Ch_Chemistry / | ||
Introductory quantum chemistry(649s).pdf |
|
Size 1.6Mb Date Dec 18, 2003 |
several semi-empirical methods is provided in Appendix F). This section also develops the Orbital Correlation Diagram concept that plays a central role in using WoodwardHoffmann rules to predict whether chemical reactions encounter symmetry-imposed barriers....
10. Introduction to Quantum Mechanics, L. Pauling and E. B. Wilson, Dover Publications, Inc., New York, N. Y. (1963)- Pauling and Wilson. 11. Modern Quantum Chemistry, A. Szabo and N. S. Ostlund, Mc Graw-Hill, New York (1989)- Szabo and Ostlund. 12. Quantum Chemistry, I. N. Levine, Prentice Hall, Englewood Cliffs, N. J. (1991)Levine. 13. Energetic Principles of Chemical Reactions, J. Simons, Jones and Bartlett, Portola Valley, Calif. (1983),...
A. Operators Each physically measurable quantity has a corresponding operator. The eigenvalues of the operator tell the values of the corresponding physical property that can be observed...
electrons and the x, y, and z (or r,θ, and φ) coordinates of the oxygen nucleus and of the two protons; a total of thirty-nine coordinates appear in Ψ. In classical mechanics, the coordinates qj and their corresponding momenta pj are functions of time. The state of the system is then described by specifying qj(t) and pj(t). In quantum mechanics, the concept that qj is known as a function of time is replaced by the concept of the probability density for finding qj at a particular value at a particular time t: |Ψ (qj,t)|2. Knowledge of the corresponding momenta as functions of time is also relinquished in quantum mechanics; again, only knowledge of the probability density for finding pj with any particular value at a particular time t remains. C. The Schrödinger Equation This equation is an eigenvalue equation for the energy or Hamiltonian operator; its eigenvalues provide the energy levels of the system 1. The Time-Dependent Equation If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrödinger equation How to extract from Ψ(qj,t) knowledge about momenta is treated below in Sec. III. A, where the structure of quantum mechanics, the use of operators and wavefunctions to make predictions and interpretations about experimental measurements, and the origin of 'uncertainty relations' such as the well known Heisenberg uncertainty condition dealing with measurements of coordinates and momenta are also treated. Before moving deeper into understanding what quantum mechanics 'means', it is useful to learn how the wavefunctions Ψ are found by applying the basic equation of quantum mechanics, the Schrödinger equation, to a few exactly soluble model problems. Knowing the solutions to these 'easy' yet chemically very relevant models will then facilitate learning more of the details about the structure of quantum mechanics because these model cases can be used as 'concrete examples'. The Schrödinger equation is a differential equation depending on time and on all of the spatial coordinates necessary to describe the system at hand (thirty-nine for the H2O example cited above). It is usually written H Ψ = i h ∂Ψ/∂t...
Σ i { - (h2/2me) ∂2/∂qi2 + 1/2 Σ j e2/ri,j - Σ a Zae2/ri,a } Ψ
+ Σ a { - (h2/2ma) ∂2/∂qa2+ 1/2 Σ b ZaZbe2/ra,b } Ψ = i h ∂Ψ/∂t. 2. The Time-Independent Equation...
equations. An analogous process must be applied to B(y) to achieve a function that vanishes at y=0: B(y) = exp(iy(2mEy/h2)1/2) - exp(-iy(2mEy/h2)1/2) . Further requiring A(x) and B(y) to vanish, respectively, at x=Lx and y=Ly, gives equations that can be obeyed only if Ex and Ey assume particular values: exp(iLx(2mEx/h2)1/2) - exp(-iLx(2mEx/h2)1/2) = 0, and exp(iLy(2mEy/h2)1/2) - exp(-iLy(2mEy/h2)1/2) = 0. These equations are equivalent to sin(Lx(2mEx/h2)1/2) = sin(Ly(2mEy/h2)1/2) = 0. Knowing that sin(θ) vanishes at θ=nπ, for n=1,2,3,..., (although the sin(nπ) function vanishes for n=0, this function vanishes for all x or y, and is therefore unacceptable because it represents zero probability density at all points in space) one concludes that the energies Ex and Ey can assume only values that obey: Lx(2mEx/h2)1/2 =nxπ , Ly(2mEy/h2)1/2 =nyπ , or Ex = nx2π 2 h2/(2mLx2), and Ey = ny2π 2 h2/(2mLy2), with n x and ny =1,2,3, ... It is important to stress that it is the imposition of boundary conditions, expressing the fact that the electron is spatially constrained, that gives rise to quantized energies. In the absence of spatial confinement, or with confinement only at x =0 or Lx or only at y =0 or Ly, quantized energies would not be realized. In this example, confinement of the electron to a finite interval along both the x and y coordinates yields energies that are quantized along both axes. If the electron were confined along one coordinate (e.g., between 0 ≤ x ≤ Lx) but not along the other (i.e., B(y)...
The two (1/2L)1/2 factors are included to guarantee that ψ is normalized: ∫ |ψ(x,y)| 2 dx dy = 1. Normalization allows |ψ(x,y)| 2 to be properly identified as a probability density for finding the electron at a point x, y. 4. Quantized Action Can Also be Used to Derive Energy Levels There is another approach that can be used to find energy levels and is especially straightforward to use for systems whose Schrödinger equations are separable. The socalled classical action (denoted S) of a particle moving with momentum p along a path leading from initial coordinate qi at initial time ti to a final coordinate qf at time tf is defined by: qf;tf S = ⌠ p•dq . ⌡ qi;ti Here, the momentum vector p contains the momenta along all coordinates of the system, and the coordinate vector q likewise contains the coordinates along all such degrees of freedom. For example, in the two-dimensional particle in a box problem considered above, q = (x, y) has two components as does p = (P x, p y) , and the action integral is: x f;yf;tf S = ⌠ (p x d x + p y dy) . ⌡ xi;yi;ti In computing such actions, it is essential to keep in mind the sign of the momentum as the particle moves from its initial to its final positions. An example will help clarify these matters. For systems such as the above particle in a box example for which the Hamiltonian is separable, the action integral decomposed into a sum of such integrals, one for each degree of freedom. In this two-dimensional example, the additivity of H:...
| © 2007 eKnigu | ||