manyelectronatoms多电子原子.docVIP

CHEM 614MODULE 9MANY ELECTRON ATOMSN12r1r2r12FIGURE 9.1With the exception of hydrogen the elements of the Periodic Table consist of atoms that have 2 or more electrons in their make up. This fact means that in our Schrdinger equation for the energy we have multiple kinetic energy operators and multiple potential energy operators to contend with. The simplest many electron atom is that of helium with a Z = 2, i.e., two electrons and one nucleon having +2 charge. Figure 9.1 is a schematic of the He atom.As with the H atom we can neglect the motion of the atom as a whole through space as contributing to the internal energy. Thus we need to account for the kinetic energy of the two electrons about the center of mass, and there are PE terms for the various Coulombic attractions and repulsions present. The hamiltonian operator in atomic units is where 1, 2, ., n identifies the spatial coordinates of each of the n electrons. Of course for He atom n = 2, and the hamiltonian becomeswhich can be written in an abbreviated formwhere the h(1) and h(2) terms are one electron hamiltonians. The third term is a two electron operator (due to the inter-repulsion of the two electrons) and it is the presence of this term that prevents us from separating the Schrdinger equation into a pair of single variable equations. Thus the 3-particle He atom is impossible to solve exactly and we need to resort to approximations methods, such as variation theory that we introduced in Module 8. A very obvious (but drastic) approximation is the one we used in our variation treatment, viz., to simply ignore the term. This is the “independent electron approximation” and it is clear that the h(1) and h(2) terms in equation (9.3) are hamiltonians for a pair of hydrogenic atoms containing a Z = 2 nucleus. The corresponding wavefunctions are the 1s, 2s, 2p, etc atomic orbitals. Ignoring the 1/r12 term in the full hamiltonian generates an approximate operatorwhere It is important to note that the 1s, 2s, etc atomic orbitals are eigenfunctions of , but not of the full hamiltonian. In this approximation (zero order) the ground state of the He atom can be written as 1s(1)1s(2), where the (1) and (2) refer to the individual electrons in the atom. As we shall see later, the He atom question can also be approached using Perturbation Theory. Are Electrons Individually Recognizable? In the macroscopic world we can easily imagine a billiard table on which are placed a set of identical billiard balls, same color and same size. The way that we can distinguish between them is by their positions relative to a fixed point on the table surface or, if they are in motion, by their individual velocity vectors. Their different trajectories make them distinguishable. Electrons are also all the same, same rest mass, same intrinsic angular momentum and same charge. There are no different colors, etc, that enable us to tag individual electrons. In the sub-microscopic world of the electron however, the momentum and position vectors fail to commute and therefore these two observables cannot be simultaneously specified with arbitrary precision. If the momentum of an electron is sharply defined, its position is very uncertain, and vice versa. Thus there is no property of an electron in an atom that we can use to distinguish it from any other electron. In a situation where there are several electrons occupying atomic orbitals we are unable to state which of the electrons occupies which orbital. Indeed, the question itself is without any meaning. The earlier description of the electronic configuration of He ground state as 1s(1)1s(2) is satisfactory because it tells us that both of the electrons are occupying the same spatial orbital designated as 1s, and there is no attempt to distinguish between the electrons. As we shall see later we can specify an excited state of He as 1s12s1, indicating one electron is occupying the 1s orbital and the other is in the 2s orbital. This description is allowable since it does not require us to state which electron is in 1s and which is in 2s. However, to write the zeroth-order wavefunction of the excited state as 1s(1)2s(2) is not allowed, since now we are specifying that electron (1) is in the 1s orbital and electron (2) is in 2s. This implies that we can somehow distinguish electron (1) from electron (2), which we cannot. Instead we must use superposition functions such as which do not specify that a particular electron is in a particular orbital. The wavefunction of a system of particles is a function of all the variables that pertain, both space and spin. Thus for electron (1) in a set of n electrons, the space variables are x1, y1 and z1 and the spin variable is s1. For the sake of brevity we denote all of these variables for electron (1) by q1, and for electron (2) by q2 and so on up to qn. Thus, the wavefunction of a system of n identical particles can be written as . Now we define the permutation or exchange operator as the operator that exchanges all the space and spin coordinates of the particles (j) and (k) in the n-particle system. At this point it is convenient to simplify our system to two electrons (e.g. the He atom), and write the function as and the exchange operator as . Thus and we see that application of the operator has been to exchange the coordinates of the two electrons. This will not affect the state of the system since the electrons are not distinguishable (an electron is an electron is an electron) since in effect it serves only to change the labels we placed on the electrons. Clearly application of the exchange operator a second time will bring the system back to the original description, so we can say where is the unit operator. It can be shown that the eigenvalues of any operator whos square is the unit operator are +1 and 1. Thus when our function is an eigenfunction of we see thatWhen the result of the operation retains the sign of the original function we say that the function is symmetric with respect to interchange of electrons (1) and (2); when the result of the operation changes the sign of the function we say that the function is antisymmetric with respect to the interchange. The conclusion applies in the general case: a wavefunction for a system of n identical particles must be symmetric or antisymmetric with respect to every possible interchange of any two of the particles. Since the particles are identical it is not conceivable that some interchanges would turn out to be symmetric and others antisymmetric. The periodic table of the elements provides abundant evidence (see later) that for systems of electrons (atomic states) only the antisymmetric result is found and we can formulate another quantum mechanical postulate that “The wavefunction of a system of electrons must be antisymmetric with respect to the interchange of any two electrons.”This is a very important postulate in Quantum theory and it is due to Wolfgang Pauli who employed relativistic quantum field theory to show that fermions (particles with half-integral spin; , and so on) the family to which electrons belong require antisymmetric wavefunctions, whereas bosons (s = 0, 1, 2, and so on) require symmetric wavefunctions. To demonstrate the effect of the Pauli principle for identical fermions we start with the recognition ( from the postulate) that the antisymmetry requirement is Now suppose we assign the same four coordinates to two (identical) electrons. Thus, x1 = x2, y1 = y2, z1 = z2, and s1 = s2, thus q2 = q1. Substituting this in equation (9.8) leads to or, 2y = 0and the probability of finding the two electrons in the orbital defined by y is zero. Thus two electrons having the same spin function have zero probability of being found having the same values of the space coordinates, in other words cannot co-exist in the same region of space. This “Pauli repulsion” forces electrons of the same spin to keep apart from one another, e.g., in different orbitals. The Helium Atom We saw above that the electronic configuration of the ground state of He in the zeroth order spatial wavefunction could be written as 1s(1)1s(2), i.e., both electrons occupying the same orbital. From the foregoing discussion, this description requires us to consider the spin state of the two electrons. At a superficial level, there are four possible spin combinations of the two electrons, thusThe principle of indistiguishability of identical particles is clearly upheld in the first pair of combinations, since both electrons are assigned the same spin function. However, this is not the case for the second two combinations since now we show a particular electron with a particular spin function, and this violates the principle. Applying to the combinations shows that the first and second are both symmetric and the third and fourth are neither symmetric nor antisymmetric with respect to interchange. These last two then are unacceptable. Moreover, even though the first and second spin combinations are acceptable in themselves, when they are coupled with the 1s(1)1s(2) spatial function, which itself is symmetric to particle interchange, we find the Pauli principle forbids the combinations and since they are overall symmetric to electron interchange thereby violating the Pauli rule. The way to proceed is to seek a spin function that is an antisymmetric superposition. The two candidates for this are given by These two functions are normalized linear combinations of. They are the symmetric (+) and antisymmetric (-) eigenfunctions of . Since we are combining one of these with the symmetric spatial function 1s(1)1s(2) to make an overall antisymmetric total function, we need the antisymmetric superposition, in which case the He ground state is given by This is the total wave function for the ground state of He (in zeroth-order) including spatial and spin contributions. Note that it is a single state; there are no degeneracies. At this point, it is worth noting that the hamiltonian operator contains no spin terms so that the energy of a state is not affected by the spin terms in the wavefunction, at least to a reasonable approximation. Now we can take a preliminary look at He in an excited electronic state, i.e. a state in which one of the 1s electrons is promoted to the n = 2 level. In principle, this new state could be described as 1s12s1 or 1s12px. However, in non-hydrogenic systems (more than a single electron) the 2s orbital is at a slightly lower energy than any of the three 2p orbitals. At first sight the two-electron combination might be written as 1s(1)2s(2) or 1s(2)2s(1), but these descriptions imply that we can identify electrons (1) and (2) as being in the designated orbitals. This clearly violates the concept of indistiguishability. Moreover, the two product functions are neither symmetric nor antisymmetric when operated upon by the exchange operator and thus cannot meaningfully contribute to a total wavefunction that must be overall antisymmetric with respect to electron exchange. In such a situation we resort to the use of a superposition, which can satisfy the symmetry requirements. One of these new wavefunctions is given by which is symmetric on electron interchange, i.e. on operating with the exchange operator. The antisymmetric superposition is The symmetric spatial combination can be an allowed state of the He 1s2s configuration if it is associated with an antisymmetric spin term, and similarly the antisymmetric spatial combination can conform to the Pauli principle if associated with a symmetric spin term. Thus, four total wavefunctions satisfy the requirements for the 1s2s excited state of He. These areThus, four distinct states can be generated under the configuration 1s2s. In order to obtain energy terms for these four possible states we need to see how the wavefunctions transform under the application of the full hamiltonian, equation (9.2). Our functions have been set up using the independent electron approximation in which we ignored the inter-electron repulsion term in equation (9.2). Therefore, the functions will not be eigenfunctions of and therefore we cannot simply compare the eigenvalues. However, we can calculate the average energies by evaluating the expectation values of with the four functions listed. ThusNote that the usual denominator is missing since both space and spin parts are normalized and thus the denominator is unity. Nothing in the operator we are using will interact with any of the spin contributions in the four functions listed, and since the spin terms are normalized, their integrals come out to be unity. Therefore, we see that the average energy will be determined by the spatial contribution, and as three of the functions contain the same spatial part, they form a three-fold degenerate set. It is important to note that the factorization into spatial and spin functions is only possible with two-electron cases. Only has a spatial part that is different and therefore we can anticipate this function to have a different energy than the three functions. From this we expect that the excited state of He that we have designated 1s2s will have two energy states, one of which is triply degenerate. In order to find what the energy values are we need to grind through some integrations and the procedure for this is laid out in Appendix I. There we find the conclusion that the two energies are given bywhere the first two terms on the RHS are the energies of the 1s and 2s eigenvalues for He+ (Z=2), respectively. The integrals J (the Coulomb integral) and K (the Exchange integral) are given byJ is always positive because it deals with the Coulombic interaction between the two electron distributions (1s*1s and 2s*2s). These distributions are everywhere negative and so the interaction is repulsive, and thus the energy of the combination is raised (less negative). The integral K has the product functions in the integrand differing by exchange of electrons, hence its name. As Appendix I explains K has negative and positive contributions but is positive overall and not as large as J. The procedure in the Appendix shows that the triply degenerate level is lower in energy than the non-degenerate level and the separation is 2K. We can think of the J term as resulting from homogeneous/time-averaged electron densities in the charge-clouds of the orbitals and the K term as being a correction arising from the inhomogeneities that occur as a consequence of electrons wanting to avoid each other. For example if we approximate the two linear combination spatial terms in equations (9.12) and (9.13) as being of the formand where a is a function of the radius vector r1 and b is a function of the radius vector r2, then as r1 approaches r2 then y- tends to vanish, i.e., the electrons tend to avoid each other in the difference combination. This leads to the Fermi hole. In a magnetic field the triply degenerate level is split into three discrete energy states-it is a triplet of states, or a triplet state.Slater Determinants In our above discussion of the He atom (ground and excited states) we have found it easy to write down the antisymmetric total wavefunctions as relatively simple multiples of spatial and spin parts. It was easy because we had to deal with just two electrons. It is another matter to construct an asymmetric wave function for N electrons, by inspection. In 1930s Slater introduced the concept of using determinants to construct the required wavefunctions. As an example let us take the He ground state wavefunction that above we have shown to beThis wavefunction is composed of a symmetric space function and an antisymmetric spin function so it is overall antisymmetric with respect to interchange of the pair of electrons-as Pauli requires. As a determinant this becomesExpanding the determinant and rearranging leads to the antisymmetric wavefunction shown above. The individual terms in the determinant are spinorbitals that describe the spatial and spin state of the appropriate electron. The determinant is formed by placing the individual spinorbitals along the rows and the associated electrons in the columns (some authors reverse this procedure, but this has no effect on the determinant since interchanging columns and rows leaves the determinant unchanged). Determinants such as that in equation (9.20) are called Slater determinants and the wavefunction in equation (9.20) is called a determinantal wavefunction. If the labels (1) and (2) are interchanged in the determinant this places electron (1) in column 2 and electron (2) in column 1, i.e. two columns of the determinant are interchanged, which has the effect of changing the sign of the determinant (see Barrante, chapter 9), and therefore of the wavefunction, hence the determinant represents an antisymmetric function. Furthermore, suppose that our two electrons in the 1s orbital have the same spin, say a. This changes the determinant as followsNow two rows (the first and second) are identical, which causes the determinant to have the value of 0 (irrespective of the order of the determinant). Thus the wavefunction for a state in which both electrons have identical sets of space and spin variables is zero and that state therefore does not exist. There are shorthand ways of writing Slater determinants. One is to indicate b-spin by a bar over the spatial orbital designator; if the bar is omitted a-spin s indicated. Thus, the wavefunction in equation (9.20) would becomeWe can use a four-electron example to illustrate the general case, where Ui represents the ith spin or