* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Indistinguishable particles, Pauli Principle, Slater
Ensemble interpretation wikipedia , lookup
Second quantization wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Molecular orbital wikipedia , lookup
Dirac equation wikipedia , lookup
Ferromagnetism wikipedia , lookup
Particle in a box wikipedia , lookup
Canonical quantization wikipedia , lookup
Probability amplitude wikipedia , lookup
Hydrogen atom wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Renormalization group wikipedia , lookup
Quantum state wikipedia , lookup
EPR paradox wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Coupled cluster wikipedia , lookup
Bell's theorem wikipedia , lookup
Electron scattering wikipedia , lookup
Hartree–Fock method wikipedia , lookup
Double-slit experiment wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Elementary particle wikipedia , lookup
Spin (physics) wikipedia , lookup
Matter wave wikipedia , lookup
Tight binding wikipedia , lookup
Atomic orbital wikipedia , lookup
Wave–particle duality wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Atomic theory wikipedia , lookup
Identical particles wikipedia , lookup
Electron configuration wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Indistinguishable Particles In classical mechanics, we can keep track of all particles just by watching them. The task may be difficult in practice but it contains no basic difficulties. This is equivalent to painting each object with a distinct color (or label). Consider a pool game in which all the balls were painted black, but you know their initial identity (2 ball, etc.). By watching the game carefully, you could keep track of the balls as though they had their original colors. But what if you left the room, and the game continued? The latter case is analogous to the quantum case: when the wavefunctions of two identical particles overlap (i.e., they are within a deBroglie wavelength of each other), it is generally not possible to retain their identity. This means that labeling the particles (1,2, . . . n) as we must do to write out the Hamiltonian has a conceptual problem. We must construct the final wavefunctions such that physical observables do not change if we exchange the identity labels on any two identical particles!! Since ψ * ψ is the observable, the wavefunction can only change by ±1 when this exchange takes place. Actually, ψ could change by eiφ , but we will find that φ is either 0 or π. Notation : ψnAm(2) means that particle 2 is in STATE n,A,m. ψ2pz(2) similarly means that particle 2 is in the 2pz orbital. Consider the ground state of helium, where both electrons are in the 1s state. We had obtained the result ψ = φ1s(1) φ1s(2). There is no problem with a 1 - 2 exchange here. However, suppose that we consider the first excited state with ψA = φ1s(1) φ2s(2). There is big trouble because 1–2 exchange gives a new function ψB = φ1s(2) φ2s(1). which is NOT the same as ψA. To satisfy indistinguishability, we can construct linear combinations of these two to obtain new functions ψa and ψs where exchange of 1-2 labels has no observable effect. 91 ψs = F 2 I φ (1) φ GH 2 JK ψa = F 2 I φ (1) φ GH 2 JK 1s 1s 2s 2s (2) + φ1s (2) φ2s (1) (2) − φ1s (2) φ2s (1) We can readily see that these two new functions are normalized eigenfunctions that leave the wave function unaffected (ψs) or multiply it by -1(ψa) when particles 1 and 2 are interchanged. Thus, these two functions now satisfy our requirement that exchanging the labels on indistinguishable particles can have no observable result. These two functions are called respectively symmetric and anti-symmetric with respect to interchange of identical particles 1 and 2. Formally these functions can be constructed using the permutation operator, P ij . This operator is defined by its function, P ij f(r1, . . .ri . . .rj . . . ) = f(r1, . . .rj . . .ri . . . ) What are the eigenvalue(s) of P ij2 ? If we include spin in the helium ground state 1s2, then we can naively construct four ground state wavefunctions, ψ1 = φ1s(1)α(1) φ1s(2) α(2) ψ2 = φ1s(1) α(1) φ1s(2) β(2) ψ3 = φ1s(1) β(1) φ1s(2) α(2) ψ4 = φ1s(1)β(1) φ1s(2) β(2) All four of these functions are solutions of the He atom Schrödinger equation developed earlier, with spin added. Functions ψ1 and ψ4 satisfy the identical particle issues with a permutation eigenvalue of +1 (symmetric). Functions ψ2 and ψ3 are not invariant upon particle permutation. In fact, P 12 ψ 2 = ψ 3 P 12 ψ 3 = ψ 2 However, we can construct linear combinations of ψ2 and ψ3 which do satisfy this requirement: 92 ψs = F 2 I [φ (1) φ (2)][α(1)β(2) + β(1)α(2)] GH 2 JK ψa = F 2 I [φ (1) φ (2)][α(1)β(2) − β(1)α(2)] GH 2 JK 1s 1s 1s 1s We now have four functions, ψ1, ψ4, ψs and ψa which all are eigenfunctions of P 12 with eigenvalues +1,+1,+1 and -1, respectively. Enter the Pauli Exclusion Principle. The Pauli Exclusion Principle imposes further requirements for an acceptable wave function. It states that The state function for a system of identical half integral spin particles must be anti-symmetric when any two are interchanged. For identical integral spin particles, the wave function must be symmetric upon interchange. Based upon the Pauli Principle, only ψa is an acceptable state function for the helium ground state. (We will soon see that an equivalent statement of the Pauli Principle for electrons is that no two electrons can have the same set of n, A, mA and ms quantum numbers.) Thus the ground state wave function for He is given by ψ1s2 = F 2 I φ (1) φ (2) α(1)β(2) − β(1)α(2) GH 2 JK 1s 1s We will now look briefly at some properties of this function. Consider the operator for the total z-component of electron spin angular momentum, S ztot = S z1 + S z2 . We have S ztot α(1)β(2) = [S z1 + S z 2][α(1)β(2)] = = = [α(1)β(2)] − [α(1)β(2)] = 0 2 2 and also S ztot β(1)α(2) = [S z1 + S z2][β(1)α(2)] = = = − [β(1)α(2)]+ [β(1)α(2)] = 0 2 2 Thus the He 1s2 wave function is an eigenfunction of S ztot with eigenvalue zero. With a little more work, we can also show that this wave function is an eigenfunction of S2tot with eigenvalue zero. It is quite straightforward to show that this wave function is also an 93 eigenfunction of the L2tot and L ztot operators with eigenvalues of zero. We describe the helium ground state with the term symbol He 1 1S0 The other three ground state wavefunctions were rejected by the Pauli principle because they represented electrons with spins parallel, and, since both electrons were in 1s orbitals, violated our intuitive notion of not putting electrons in the same orbital when they have the same spin orientation. The first excited state of He is 1s 2s and we do not expect to have this problem. We should be able to put α electrons in both orbitals, etc. In fact, a procedure just like that completed above yields for the 1s 2s state a total of four wavefunctions which satisfy the Pauli antisymmetry requirement, F 2 I φ (1) φ (2) + φ (2) φ (1) [α(1)β(2) − β(1)α(2)] GH 2 JK F 2 I φ (1) φ (2) − φ (2) φ (1) [α(1)α(2)] = G H 2 JK F 2 I φ (1) φ (2) − φ (2) φ (1) [α(1)β(2) + β(1)α(2)] = G H 2 JK F 2 I φ (1) φ (2) − φ (2) φ (1) [β(1)β(2)] = G H 2 JK ψ1 = ψ2 ψ3 ψ4 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s Note how all four functions are antisymmetric. The first one is just like our ground state He wave function, with one of 1s orbitals replaced with 2s and a little mathematical gymnastics exercised. Thus ψ1 is an eigenfunction of S z tot with eigenvalue zero. As earlier, we can show that this wave function is an eigenfunction of S2tot with eigenvalue zero, and also an eigenfunction of the L2tot and L ztot operators with eigenvalues of zero. Thus ψ1 can be represented as He 2 1S0. We thus are beginning to characterize the He atom by joint properties of the electrons, rather than the properties of individual electrons. We have described the helium ground state with the term symbol He 1 1S0 A physical picture of the angular momentum relationships between the two electrons in this singlet (S=0) state is emerging. 94 In the singlet (S=0) state, the two electrons have angular momentum vectors which lie at an indeterminate position on the cones. However, given the position of one, the other points in the opposite direction, giving a net of zero. More on these later . . . The other three ground state wavefunctions were rejected by the Pauli principle because they represented electrons with spins parallel, and, since both electrons were in 1s orbitals, violated our intuitive notion of not putting electrons in the same orbital when they have the same spin orientation. Functions ψ2 and ψ4 admit of a simple physical picture, and correspond to the two electrons both having spin up or both having spin down. These functions are eigenfunctions of S z tot with eigenvalues ±= respectively. These wavefunctions are eigenfunctions of S2tot with eigenvalue 2=2, and also eigenfunctions of the L2tot and L ztot operators with eigenvalues of zero. The final function ψ3 is an eigenfunction of S ztot with eigenvalue 0, and also an eigenfunction of S2tot with eigenvalue 2 =2. We recognize these as the three components of the lowest excited triplet state of He, He 2 3S. A simple physical picture of these three triplet spin states follows. 95 At this level of sophistication, all four of these 1s 2s excited states have the same total energy. In fact, they do not and the 2 3S states lie almost 50 kcal/mol below the 2 1S state. We also know that all of these calculations that neglect the electron-electron repulsion are too crude to produce numbers of chemical relevance. Thus, we must introduce approximation methods. The two most important methods are known as Perturbation Theory and Variational Approaches. The variational approach is the basis of many large scale computations, and is the one we describe. However, first, we need to find a more general way to write out wavefunctions that satisfy the Pauli Principle. This can be done elegantly and generally by the use of Slater Determinants. Slater Determinants, Spin and Multielectron Atoms We have seen that linear combinations of orbital functions are generally required to satisfy both the identical particle issues and the overall antisymmetry of the wave function required by the Pauli Principle. We obtained the four possible 1s 2s wave functions for He 96 and interpreted them as one singlet (spin paired) function and three triplet (spins parallel or unpaired) functions. For many electrons, this ad hoc construction procedure would obviously become unwieldy. However, there is an elegant way to construct an antisymmetric wave function for a system of N identical particles. It was devised by John Slater, and makes use of the fact that the sign of a determinant reverses upon interchanging any two columns or rows. Thus if we construct a determinant form of the wave function where the columns represent different particles, then we are guaranteed that the wave function is antisymmetric with respect to interchange of any two electrons. Let us look at the three electron atom Li for an illustration. We must place the three electrons into three spin-orbitals. Let us choose 1sα, 1sβ and 2sα. We now construct a determinant where these spin orbitals are listed in rows and we assign each particle (1, 2 or 3) to a column. We thus have the determinant wave function, 1sα(1) 1sα(2) 1sα(3) ψ Li = 1 3! 1sβ(1) 1sβ(2) 1sβ(3) 2sα(1) 2sα(2) 2sα(3) The individual spin orbitals are each normalized. Since an n x n determinant gives n! terms, the leading coefficient serves to normalize the wave function. Note that the cross terms in ψLi* ψLi will vanish in the integration since each will contain one orthogonal part. We obviously could also have constructed a second ground state wave function for Li using the 2sβ spin orbital. In general, it may be necessary to employ a linear combination of Slater determinants to obtain the desired wave function. All of these functions are degenerate as the present level of approximation, but when we begin to account for the 1/rij terms, they will have different energies. Finally, we note that the value of a determinant with two identical rows or columns is ZERO. Upon inspection of the Li determinant above, we see that if two of the particles are in the same spin orbital, then two rows of the determinant are identical. Since having two particles in the same spin orbital is another way of saying that they have the same quantum numbers, this result leads to the more familiar statement of the Pauli Principle: No two electrons in an atom can have identical quantum numbers. At this stage, we have the ability to construct the periodic table. With the method to construct antisymmetric wave functions in hand, we can go on to obtain the Hartree Fock limiting form and add in electron correlation. We must still worry about obtaining the total angular momentum and spin for the atom. To begin this process, we first revisit the He 1s2 and 1s 2s functions that were shown earlier to be singlet and triplet functions: The Slater Determinant for the He ground state is ψ He = 1 2! 1sα(1) 1sα(2) 1sβ(1) 1sβ(2) 97 exactly our ground state singlet. This is a closed shell system, and we could only find one Slater determinant. When the system is open shell, as the 1s 2s state, there are a number of Slater determinants (4 in this case). ψ1 = 1 2! ψ2 = 1 2! ψ3 = 1 2! ψ4 = 1 2! 1sα(1) 1sα(2) 2sβ(1) 2sβ(2) 1sα(1) 1sα(2) 2sα(1) 2sα(2) 1sβ(1) 1sβ(2) 2sβ(1) 2sβ(2) 1sβ(1) 1sβ(2) 2sα(1) 2sα(2) We see that the wave functions that are eigenfunctions of total spin and total orbital angular momentum in the open shell case are in general linear combinations of Slater determinants. Now we are ready to proceed in our investigation of approximation methods. 98