Diatomics in Molecules - DIM

Transcription

1 Chapter 4 Diatomics in Molecules - DIM In this chapter it is shown how the Diatomic in molecules method is applied to systems of dihalogens in a rare gas environment. irst, the construction of the amiltonian is shown and in the subsequent section each part is discussed in detail, including derivation and application of the particular elements. The proper choice of the basis for the amiltonian is discussed, which is chosen in this work to consist of the Valence-Bond wave functions of the guest molecule. To this, all the other interactions are added, the halogen-argon interaction and the spin-orbit coupling. ere electron-hole dualism has to be taken into account, where the electron hole of the halogen s p-orbital is treated like a single electron, which will lead to a modification of the respective coupling elements. 4.1 otentials for XY in Argon The potentials for dihalogens in Argon were calculated using the Diatomics in Molecules (DIM) approach [7, 8, 9, 11, 73, 3, 75, 9, 39,, 1, 13, 14]. ere the total amiltonian of the system is built by taking into account all of the interactions of the system. These are the following: 1. interaction between the closed shell rare gas atoms, which are described by pair potentials. interaction between the atoms of the molecule, X and Y, including spin-orbit interaction 3. interaction between the open shell halogen atoms of the molecule and the argon atoms, respectively 5

2 " Theoretical Concepts: Diatomics in Molecules Approach 51 rom this the total amiltonian of the system results: $ ! $&%(')* + "-,." + "-,." + ",." " + "8,.9;:=< >? :;<@A9;:=<CB :=<D9;:;< (4.1) To develop the amiltonian, one has to accomplish the following steps 1. choose a basis for the amiltonian usually the molecular amiltonian in 9G I K -representation for each halogen atom is taken as basis. This has the advantage that the resulting matrix is diagonal with the eigenvalues of the free molecule (without spinorbit coupling) as diagonal elements. The amiltonian becomes an LNM>L matrix where L equals the number of the wavefunctions. In the present case the amiltonian will consist of 3 wavefunctions and thus built a 3x3 matrix (see 1.1.3).. to this matrix all the other interactions are added and then by diagonalizing the eigenvalues of the system are achieved. 4. Wavefunctions The strategy is to build first the wavefunctions of the free molecule without spin orbit coupling and then to add the coupling separately. In this chapter the interhalogen Cl is taken as example for systems XY to show the general derivations of the method. The Valence Bond approach is used for creation of the wavefunctions. ere all the core electrons of the Cl and atoms are ignored and only the valence electrons are taken into account.the corresponding configuration is QQTU7TVGW for the atom and XQTU7TVIYTZ$7AZ[GW for the Cl atom. The hole in the electronic configuration is equivalent, and therefore, treated like a single electron. This corresponds to the particle-hole dualism and leads to equivalent wave functions for the molecule. The only problem arises by treating the spin-orbit coupling of an electron configuration like W as one like problem will be discussed later in ection 4.4.., but this

3 Construction of the VB-wavefunctions The electronic configuration of the and Cl-atoms are, respectively: Cl X7TU7TTGW X7T3QAVGYAZ$7AZVGW ach atom has an angular momentum of l=1 with m =-1,,1 and a spin of. This leads to six possible combinations for spin and angular momentum for each atom. or the Cl molecule we get the possible projections for the total angular momentum: To get a complete set of all possible combinations, we then will have 3 wavefunctions, thus a 3x3 matrix for the amiltonian. Construction of these wavefunctions is performed using ladder-operators for angular momentum and spin. irst we have to determine good quantum numbers, which will only appear for commuting operators: = = and cyclic = = and cyclic = = The good quantum numbers without spin-orbit coupling are then L, M, and M

4 9 Theoretical Concepts: Diatomics in Molecules Approach 53 With and cyclic, the ladder operators are defined as [1]: (4.) (4.3) these operators can be used to construct new wavefunctions: " * " * (4.4) 9 " * " * 9G (4.5) and the same for the spin: (4.) 9 9G (4.7) Before applying these ladder operators, it is helpful to build a matrix consisting of the possible values of and and there tabulate all existing configurations:

5 54 M , , , , , M , , , , where e.g. denotes the later-determinant with quantum numbers L=, M =, =1, M =1. Then one unique configuration is taken (like L=, =1, M =, M =1) and the ladder-operators are applied. or example, to get the configuration L=, =1, M =1, M =1 the lowering operator 9 is applied. Next, operating with the lowering operator on the configuration L=, M=1, =1, =1 will lead to the next configuration L=, M=, =1, =1. By applying the raising and lowering ladder-operators for the spin the other columns are created.to make sure that the resulting wavefunctions are orthogonal to each other, the chmidt algorithm for orthogonalization is taken into account. inally these procedure is worked out until all configurations are determined. or many-electron diatomic molecules, commutes with the operator for the axial component of L, which has the possible values Thus for characterization of the appropriate states for =, 1,, respectively [1]. with is used..g..

6 Theoretical Concepts: Diatomics in Molecules Approach xample We start with a unique configuration: " * " " 9 9 Ö Q Ö Using the later determinants this can be written as shown below, where in our case index a belongs to the fluorine atom and index b belongs to the chlorine atom: 9 D 5 D 3 3 U D 5

7 5 patial part of wf (first) (second) 9 (first) (second) (first) (second) (third) (first) (second) 5 D 9 D 9 D 9 D T D 9 T T T 9 T 9 9 V A 9 T 9 V 9 T T 9 A A 9 9 T 5 V 9 5 T 9 5 A 9 5 A 5 T 9 5 T D A A 5 D T T 9 D D VT 9 D D A A 9 D 5 V T A A D T A 9 D T T T A T T (singlet) (singlet) (singlet) -1 (singlet) 1 (singlet) -1 (singlet) 1 (singlet) - (singlet) (singlet) -1,,1 (triplet) -1,,1 (triplet) -1,,1 (triplet) -1-1,,1 (triplet) 1-1,,1 (triplet) -1-1,,1 (triplet) 1-1,,1 (triplet) - -1,,1 (triplet) -1,,1 (triplet) Table 4.1: Valence-bond wavefunctions of dihalogens These wavefunctions have the general form (4.8) The application of the general form in eq. 4.8 on the wavefunctions is shown in

8 Theoretical Concepts: Diatomics in Molecules Approach 57 two examples: 5 A A A A 5 T D T V A A T D T A 5 A A 4.3 Interaction with Argon igure 4.1: alogen-argon interaction To describe the interaction of the closed shell atoms and the open shell atoms first order perturbation theory is used [], introducing the perturbation of the atomic energy levels of by rare-gas neighbours. All core electrons of the halogens are assumed to be unaffected by the rare gas atoms. The interaction is calculated regarding the hole in the electron configuration of the halogen atom. The resulting amiltonian for this system is: (7 With r being the position vector of the electron hole and V(r, (4.9) ) the interac- tion potential between halogen atom and a rare gas atom with position vector,

9 58 where the sum over all argon atoms is built. The basis of the DIM-amiltonian consists of wavefunctions of the type which are arranged in blocks with same spin. (4.1) with a=, b=cl and i=ar. urther, the interaction potential 9 is expanded in Legendre polynomials: where 9 at r (see ig. 3 (7 I Q 7 (4.11) denotes the angle between atom k and the electron(hole) centered 4.1).This orientation angle is thus, within the DIM-model, the only electronic degree of freedom [79]. or a system like a halogen atom l=1. o for the spherical harmonics all odd terms will vanish by integration and all terms higher L= won t appear due to l=1, see below. The present basis for the electron(hole) will be " + 9G " " C, " B U 9 B "B (Q " B Q (7 (7 (7 Q " B (7 " B " B * " * "! " * (7 Q 7 3 Q (4.1) "B " B (4.13) (4.14) Generally, as shown e.g. by Apkarian and chwentner [19] the matrix elements are: B "B I " "B * 9 * " B (Q $! * 3 7 (4.15)

10 Z Theoretical Concepts: Diatomics in Molecules Approach 59 where l=1 for the p-orbitals. The quantum numbers for the matrix element "B * (Clebsch-Gordan coefficients) satisfy the triangle relation K " B (4.1) (4.17) (4.18) or l =l =1, therefore, only the terms L= and L= contribute to the sum I V and V describe the isotropic and anisotropic part of the potential,respectively, and can be represented by the pair potentials and [3, 5]. (Q (Q 4.4 pin-rbit Coupling Z (7 (7 (Q5 (7D (4.19) (4.) pin-rbit coupling is an important factor for diatomic molecules, both for the energetics and for the dynamics of the system. or dihalogens in general there will appear the problem of the particle-hole dualism, where the hole in the electronic configuration will lead to the same spin-orbit coupling term as a single electron but with opposite sign for the values of the spin-orbit coupling, as referred to in section To give a consistent presentation, first the principles of spin-orbit coupling will be discussed. The given results will then be used for the system Cl. A long, straight conductor generates a magnetic field whose flux lines lie perpendicular to the flow of charge and build a right hand oriented system. This is described by araday s rule: M (4.1) with velocity. The interaction of the dipole moment associated with the spin of the electron with this magnetic field is given by (4.)

11 igure 4.: Magnetic field generated by a straight conductor with being the gyromagnetic constant derived by Dirac s relativistic treatment of the electron and has a value of, and means the electronic spin angular momentum. Likewise, for a nuclear spin the spin-orbit coupling is given by: ) ) There are several interactions that have to be discussed: ) (4.3) electron spin - nuclear motion electron spin - other electron motion nuclear spin - electron motion nuclear spin - nuclear motion irst we derive the expression for the first interaction. rom these, the expressions for the other interactions follow by analogy: electron spin - nuclear motion irst we have to regard that the velocity and angular momentum of the nucleus corresponds to the negative velocity and angular momentum of the electron, taking the nucleus or the electron as point of view, respectively. or slow motion Coulomb s law can be applied: IV (4.4)

12 M M Theoretical Concepts: Diatomics in Molecules Approach 1 igure 4.3: Velocity and angular momentum, taking the nucleus or electron as a point of view with o the magnetic field can be derived as: M M M (4.5) (4.) (4.7) with M (4.8) (4.9) (4.3) The dipole of the electron due to the spin has potential energy of orientation

13 : : in this magnetic field given by: By analogy, the other interactions are given as follows: electron spin - other electron motion nuclear spin - electron motion ) This term can be neglected because of the dependence on *. nuclear spin - nuclear motion ) (4.31) (4.3) (4.33) (4.34) Likewise this term can be neglected because of the dependence on *. o the total interaction can be written as 1 (4.35)

14 Theoretical Concepts: Diatomics in Molecules Approach 3 with (4.3) As a consequence, the pin-rbit coupling is split into three parts: " " " " (4.37) igure 4.4: Distances between atom a and atom b with electron i,j, respectively. where V " whereas V " and V " are the spin-orbit interactions within atoms a and b, " depends on the distance between atoms a and b. Asymptotically,,,, thus " <. The asymptotic behaviour of V is illustrated in ig V ( r ) V,a + V,b r igure 4.5: Asymptotic behaviour of the spin-orbit coupling V. Neglecting the interaction between electrons and nuclei located at different atoms, the spin-orbit coupling reduces to

15 4 This is an approximation, since for values of < (4.38) there must be a correction term proportional. Neglecting this dependence, the pin-rbit coupling depends on just the contributions from the individual atoms []: " " (4.39) (4.4) with (4.41) These so called spin-orbit coupling constant are atom-specific constants which can be measured spectroscopically. The -constants for (.51 ev) and Cl (.19 ev) are taken from [, ] epresentation of pin-orbit coupling in the basis of the model amiltonian To add spin-orbit interactions to the amiltonian, be presented in the basis p9g, p, p located at halogen atoms a () or b (Cl). has to The matrix elements for the atomic wavefunctions are derived as follows, starting with the matrix representing : 9G 9G 9G G 9G (4.4) G 9G 9G G G G I

16 Theoretical Concepts: Diatomics in Molecules Approach 5 Using " B " B (4.43) the matrix elements can be worked out: 9G 9G G G 9G 9G G I (4.44) (4.45) (4.4) All the mixed terms vanish because of orthogonality, for example: G 9G and so we get for the presentation of G 9G in the basis p9g, p, p : or the terms and, and are worked out by ladder-operators ([1, 15], see chapter 4.): and therefore: (4.47) and similarly and for the spinfunction Q X 9G 9G 9 9 (4.48) (4.49) (4.5) (4.51) (4.5) 9G 9G Q (4.53)

17 Using the relations: " B 9G " B 5 " B (4.54) " B9G (4.55) for ladder operators, the matrix elements of and, e.g., are evaluated as follows: G I I G I 9 I G 9 I G I I G 9G G G G 9 G 9G A second example gives a value for the matrix element: G 9G 9G G 9G 9 9G G 9G I G G G 9G G 9 9G I 9

18 Theoretical Concepts: Diatomics in Molecules Approach 7 so we finally arrive at the representation of l and l in p9g, p, p basis: (4.5) (4.57) The same procedure applied to the spin yields and in terms of ladderoperators s and s9 : Q Q 9 or these ladder-operators the same relations as for the spatial part are valid:, thus Q " 9 " 9 (4.58) (4.59) " (4.) " 9G (4.1) Q 9

19 8 Q 9 o for s we get: X 9 X 9 X Likewise, we get the following relations: and thus we get the spin matrices in the basis, : (4.) (4.3) (4.4) (4.5) (4.) (4.7)

20 Theoretical Concepts: Diatomics in Molecules Approach 9 or, we use " " (4.8) Therefore, (4.9) Using these results, we proceed to the representation of C in the basis, p, p9g,p, p. or the matrix elements of, for p9g example:, p and yield

21 7 (4.7)

22 Theoretical Concepts: Diatomics in Molecules Approach 71 (4.71) (4.7) Likewise, from (4.73) (4.74) we obtain (4.75) imilarly, (4.7) (4.77)

23 7 yield inally, from (4.78) (4.79) we obtain the complete matrix for spin-orbit coupling in the basis p9g p9g,p, p : Application to the Molecular VB-Wavefunctions, p, p, (4.8) Because the basis for the model amiltonian consists of the 3 molecular VBwavefunctions (eqn. 4.1, table 4.1, section 4.), we have to apply the matrices for the atomic wavefunctions to the molecular wavefunctions to get a representation that can be added to the system s amiltonian. or this purpose we use the relation derived in section 4.4: with denoting the atom and the Cl atom. (4.81)

24 Theoretical Concepts: Diatomics in Molecules Approach 73 and (4.8) As can be seen from equation 4.8 and table 4.1, the molecular VB-wavefunctions are expressed in terms of atomic wavefunctions, as follows: (4.83) (4.84) The resulting spin-orbit matrix elements for a specific electron ( 1 ) is, for example: B T [ < As a consequence, the total spin-orbit matrix element is, in general, (4.85) xample As an explicit example the matrix element

25 74 with D T T A A T T T A T D D A A D A D5 A T D 5 is evaluated as follows, irst is only operating on the part of each expression which belongs to electron (1). yield 9G rom this we can see that will give 9 gives VG I 9G G 9G G 9G G operating for example on V 9. imilarly, operating on imilar operations on all atomic basis functions which sontribute to 9 A V T A T

26 Theoretical Concepts: Diatomics in Molecules Approach 75 inally, the A matrix element of A T A A A A A A A A A A 5 on is evaluated as A A V T A A A T T A A T T A A A T T T A A A T V A A A T T T A T T A T A A T D

27 9 9 9 L 7 imilar to this procedure we get for o for the total matrix element: A All 3x3 matrix elements are calculated by means of the same technique and the resulting values are added as off-diagonal elements to the system amiltonian. A 4.4. lectron-ole Dualism To show that this approach is valid, one has to prove that a shell occupied by n electrons, leads to the same (L,)-terms like a shell occupied with (g-n) holes with g corresponding to the degeneracy. o the configuration with n electrons is equivalent to a configuration with (g-n) occupied holes. L L L NL W or each -microstate for n electrons there is one - microstate with (g-n) holes. or example, by comparing the and configuration of an atom, in the former there is one microstate with equals to +1 and equals to + for the one hole. Let us denote this state as with the digit corresponding to the value of and the index corresponding to spin. If we would have GW configuration, there will be one configuration where the hole will be located at equals to +1 and equals to + and the five electrons inhabiting the microstates microstate with equals -1 and and, which leads to the corresponding. This will be the case for all of the

28 Theoretical Concepts: Diatomics in Molecules Approach 77 microstates of the configuration. A -term consists of ( )-states with N so for each ( )-electronstate one ( )-holestate will be included. o we can conclude that a state with n electrons leads to the same (L,)-terms as a state with (g-n)- electrons, only with a change in sign for the coupling elements.