Lecture 4: First principle calculation of the EPR spin Hamiltonian parameters including Zeeman, Hyperfine, Zero Field Splitting and Electric Field Gradients A brief review of EPR spectroscopy 1) Isotropic EPR Spectra of a paramagnetic molecule in solution (S=1/2) Consider a paramagnetic molecule (S = 1/2) with n nuclei having the following nuclear spins respectively: I 1 , I 2 , I 3 , ..., I n The effective spin hamiltonian is thus: ) H = g"H ) S z + A # # =1 n $ ) S z % ) I z # where g is the gyromagnetic factor of a paramagnetic electron, β is Bohr’s magneton for the electron, H is the magnetic field strength (supposedly parallel to the z axis), ) S z the z component of the effective spin operator of the electron, A ν is the hyperfine coupling constant of nucleus ν and ) I z " z la component of the nuclear spin operator for nucleus ν. The spin states are: M S ; M I 1 , M I 2 , ... , M I n where M S =± 1 2 et M I " = #I " , # I " + 1, ..., + I " The 1 st order energies are thus: " M S ; M I 1 , M I 2 , ... , M I n ( ) = M S g#H + A $ M I $ $ =1 n % & ’ ( ) * + The allowed EPR transitions (ΔM S = 1 et ΔM I = 0) have the following resonance conditions: h" = g#H + A " M I " " =1 n $ working at constant radio frequency v, one observes lines at the following field strengths: HM I 1 , M I 2 , ..., M I n ( ) = h" g# $ A " g# M I " " =1 n % for all possible combinations of M I " = #I " , # I " + 1, ..., + I " ; ν = 1, 2, ..., n.
Transcript
Lecture 4: First principle calculation of the EPR spin Hamiltonian parameters including Zeeman, Hyperfine, Zero Field Splitting and Electric Field Gradients A brief review of EPR spectroscopy 1) Isotropic EPR Spectra of a paramagnetic molecule in solution (S=1/2) Consider a paramagnetic molecule (S = 1/2) with n nuclei having the following nuclear spins respectively: I1, I2, I3, ..., In The effective spin hamiltonian is thus:
!
) H = g"H
) S z + A#
#=1
n
$) S z %
) I z#
where g is the gyromagnetic factor of a paramagnetic electron, β is Bohr’s magneton for the electron, H is the magnetic field strength (supposedly parallel to the z axis),
!
) S z the z component of
the effective spin operator of the electron, Aν is the hyperfine coupling constant of nucleus ν and
!
) I z"
z la component of the nuclear spin operator for nucleus ν. The spin states are:
!
MS;MI1, MI2
, ... , MIn
where
!
MS = ±1
2 et
!
MI"
= #I" , # I" +1, ..., + I"
The 1st order energies are thus:
!
" MS;MI1, MI2
, ... , MIn( ) = MS g#H + A$MI$$=1
n
%& ' (
) * +
The allowed EPR transitions (ΔMS = 1 et ΔMI = 0) have the following resonance conditions:
!
h" = g#H + A"MI""=1
n
$
working at constant radio frequency v, one observes lines at the following field strengths:
!
H MI1, MI2
, ... , MIn( ) =h"
g#$
A"
g#MI"
"=1
n
%
for all possible combinations of
!
MI"
= #I" , # I" +1, ..., + I" ; ν = 1, 2, ..., n.
To calculate the position and the intensities of all these lines, it is useful to group the nuclei with a same hyperfine constant and an identical nuclear spin (occupying generally equivalent symmetry positions) in a sub-set of equivalent nuclei. Consider e.g. par a radical with 2 equivalent protons i.e. a1 = a2 = a . La line position does depend upon the total spin M = (m1 + m2) as:
field m1 + m2 Spin state H* - a 1 αα H* 0 αβ, βα H* + a -1 ββ
where H* is the resonance field (=
!
h"
g#) and one observes thus 3 lines with the relative intensities of
1 : 2 : 1. In practice one defines le nuclear spin of a group of equivalent nuclei as
!
I" = n" # i" where iν is the individual nuclear spin and nν the number of equivalent nuclei of group ν. The degeneracies of the spin states are obtained through a procedure which is based on a scheme comparable to Pascal’s triangle. The MATLAB function: CHUCK listed below permits to obtain the intensities AMP(l) and the spin states SP(l,ν), l = 1, 2, number of lines
!
= 2I1 +1( ) " 2I2 +1( ) " ..." 2In +1( ){ } ; for an arbitrary radical with NU(ν) equivalent nuclei of nuclear spin S(ν); ν = 1, 2, ..., n. function [AMP, SP]=chuck(NU,S) % % PASCAL TRIANGLE METHOD % NDUMMY=size(NU);NG=NDUMMY(1); %D=zeros(100,2);DS=zeros(100,1);SS=zeros(size(S));NF=zeros(size(NU)); AMPM=1;KK=0;L=0;LIN=1; for I=1:NG, K=2;M=1;NPASS=NU(I);SS(I)=NPASS*S(I);NLIN=SS(I)*2+1;LIN=LIN*NLIN; NSTAR=2*S(I);NBAS=NSTAR+1;JDO=NSTAR+NBAS; for J=1:NSTAR, for JJ=1:2, D(J,JJ)=0;J1=J+NBAS+NSTAR;D(J1,JJ)=0; end end J1=J+NBAS+NSTAR; for J=1:NBAS J1=J+NSTAR;D(J1,M)=1; end if NPASS>1, for J=2:NPASS, JS=1;JE=NBAS;JDO=J*NSTAR+NBAS; for JJ=NBAS:JDO, D(JJ,K)=0; for JJJ=JS:JE D(JJ,K)=D(JJ,K)+D(JJJ,M); end JS=JS+1;JE=JE+1; end for JJ=1:NSTAR, J1=JDO+JJ; D(J1,K)=0; end
K=2/K;M=2/M; end end if I>1, II=I-1;KK=NF(II); end NF(I)=KK+NLIN; for J=NBAS:JDO, KK=KK+1;DS(KK)=D(J,M); end NBJD=NBAS+(JDO-NBAS)/2;AMPM=AMPM*D(NBJD,M); % Degeneracies %D(NBAS:JDO,M)' end %C SP=zeros(LIN,NG); for I=1:NG, NA(I)=NF(I);SW(I)=SS(I); end %FAK=1/AMPM; % % CALCULATION OF E(L), AMP(L) AND SP(L,J) % ID=find(abs(SS+SW)>0.001); %while ID~=[], while max(size(ID))~=0, JJ=1; for I=1:NG, NZ=NA(I); JJ=JJ*DS(NZ); end L=L+1;AMP(L)=JJ; % AMP(L)=JJ*FAK SP(L,:)=SW(:)'; % % for I=1:ID(1); % SW(I)=SS(I);NA(I)=NF(I); % end SW(1:ID(1)-1)=SS(1:ID(1)-1);NA(1:ID(1)-1)=NF(1:ID(1)-1); SW(ID(1))=SW(ID(1))-1;NA(ID(1))=NA(ID(1))-1; ID=find(abs(SS+SW)>0.001); end JJ=1; for I=1:NG, NZ=NA(I); JJ=JJ*DS(NZ); end L=L+1;AMP(L)=JJ; % AMP(L)=JJ*FAK SP(L,:)=SW(:)'; AMP=AMP'; The function CHUCK is illustrated in the example below. Consider the following radical :
!
[R " N = CH "CH = N " R]#
The input is thus :
nucleus ν NU(ν) = number of equivalent nuclei S(ν) = iν H 1 2 1/2 N 2 2 1
To conclude a simulation one just has to replace each line through an adequate line shape function i.e. generally a Lorentz function:
!
y H( ) =AMP j( )
1+ 4H "H j
W
#
$ %
&
' (
2
where H is the scanned magnetic field and W the line width. To obtain the simulated spectrum, all one has to do is to sum over all lines. For the example above: aH = 4.3 Gauss aN = 5.6 Gauss W = 0.4 Gauss one obtains:
Fig. line spectrum
Fig. Absorption Spectrum
Fig. Derivative of the Absorption spectrum
2) Dynamical EPR Spectrum in solution (chemical exchange) According to the theory of the spin density matrix [J. Chem. Educ. 50 (1973) 477] the intensity I(ν) of a NMR or EPR absorption at a frequency ν or a field H is given through the expression below:
!
I(") = #Cte $ Im
v 1 $U( )
µU#1 $
v P ( )
µ
%µ # i2&"µ=1
nC
'
(
) *
+ *
,
- *
. *
Where: Pj = fractional population on site j (Σ Pj = 1),
!
"µ and
!
r U µ are the eigenvalues and
eigenvectors of the density matrix A respectively:
A =
-!1 0...
0 -!N
+
-1/"1 pkj/"k...
pjk/"j -1/"N
αj = 2πi (ν-νj) - πΔν1/2 ; (νj is the resonance frequency, Δν1/2 is the width of line j and i=√-1 ) τj = the average life time on site j pjk = fraction of exchanges starting on site j terminate on site k Remark: If j ≠ k , Ajk is the 1st order exchange constant for (site j) → (site k)
A MATLAB procedure to simulate a dynamical EPR spectrum RPE (Dynesr) is listed below: %--------- % Dynesr %--------- % % Notes: - different configurations can have different g-values, % this is expressed in the shift % - the units of shift, rate, line width, and hfs contants % are mm. (i.e. Gauss or 1/sec.; since h*nu = g*beta*H ) % % dynamical parameters: Each line of matrix data_dyn corresponds % ==================== to a different configuration (nC in all) % % 1st column: peak-to-peak widths % % 2nd column: shift of resonance frequency due to % delta(g-value) % % 3rd to (2+nC)-th columns: populations on the diagonal part, % rate constants on the lower triang. part % data_dyn=[8.0 0.5 14. 0. 8.0 0.5 28. 10.]; ndummy=size(data_dyn);nc=ndummy(1); w=data_dyn(:,1);shift=data_dyn(:,2);r=data_dyn(:,3:nc+2);p=diag(r); for mu=2:nc mum=mu-1; for la=1:mum r(la,mu)=r(mu,la)*p(mu)/p(la); end end % % HFS parameters: Each line of matrix data_dyn corresponds % ============== to a different group of equivalent nuclei % % 1st column: number of nuclei in group % of equivalent nuclei % % 2nd column: nuclear spin, % % 3rd to (2+nC)-th columns: HFS coupling constant % for each configuration % data_hfs=[1 0.5 189. 161. 1 0.5 8. 6.5 1 0.5 21.5 20. 1 0.5 17. 11.5]; ndummy=size(data_hfs);na=ndummy(1); nu=data_hfs(:,1);s=data_hfs(:,2);a=data_hfs(:,3:nc+2); % [amp, sp]=chuck(nu,s);nl=length(amp); x=sp*a;xmn=min(min(x)')-10*max(w);xmx=max(max(x)')+10*max(w); % for mu=1:nc rr=0; for la=1:nc if mu ~= la rr=rr+r(mu,la); b(mu,la)=r(la,mu)+i*0; end end db(mu)=-rr-w(mu); end %
for j=1:nl % get b for mu=1:nc x(j,mu)=x(j,mu)+shift(mu); b(mu,mu)=db(mu)-i*x(j,mu); end % get eigenvalues, eigenvectors and its inverse [u,d]=eig(b);ui=inv(u); % us=ones(1,nc)*u;up=ui*p; lambda(j,:)=diag(d)'; sigma(j,:)=amp(j)*us.*up'; % % position: -imag{lambda(j,mu)} % intensity: real{sigma(j,mu)} % w(relaxation): -imag{lambda(j,mu)}/{sqrt(3)/2} % w(exchange): -real{sigma(j,mu)} % end % % convolute spectrum y = sum { sigma(j,mu)/(lambda(j,mu)+) } % j=1:nl % mu=1:nc % dfr=0.1*min(w);npt=1+(xmx-xmn)/dfr; for k=1:npt fr=xmn+dfr*(k-1);z=0+i*0; for j=1:nl for mu=1:nc xi=lambda(j,mu)+i*fr; % Absorption %z=z+sigma(j,mu)/xi; % First derivative of the absorption z=z+sigma(j,mu)/(xi*xi); end end y(k)=-imag(z); end plot([xmn:dfr:xmx],y) The above script contains the parameters for the example below i.e. the dynamical exchange of the ascorbic acid radical with Ca2+. The result of this simulation is represented in the Fig. below:
Fig. Dynamical EPR spectrum Ca -ascorbate(.)
3) Anisotropic EPR spectra of a mono-crystal and of polycrystalline samples The effective spin hamiltonian for a paramagnetic mono-crystal (Zeeman and hyperfine interactions) reads:
!
) H
spin = ") S # g #
r H ( ) +
) S #A #
) I
where β is Bohr's magneton,
!
) S is the effective electron spin operator, g is the tensor of the Zeeman
interaction,
!
r H is the applied magnetic field, A is the tensor of the hyperfine interaction et
!
) I is the
nuclear spin operator. For a given electron spin S and a given nuclear spin I, a 2nd order perturbation treatment yields the following expressions for the Zeeman energy EZe and the hyperfine energy EHF respectively: EZe = g!HMS ,
EHF = KMSMI +1
2g!HL2 "K2[ ]MS +
1
2Tr A2( ) " L2[ ] I I +1( ) "MI
2[ ]MS " 1Kdet A( ) S S +1( ) " MS
2[ ]MI# $ %
& ' (
où:
g2 =r e † !g2 !
r e ,
g2K2 =r e † ! g† !A2 !g !
r e ,
g2K
2L
2=
r e
† ! g† !A4 !g !r e ,
r e =
sin" cos#
sin" sin#
cos"
$
%
& &
'
(
) )
From these expressions one obtains resonance fields:
!
H MS " MS #1, MI , S, I( ) =h$
g%
HF 1st order
!
"1
g#KMI
HF 2nd order !
1
2 g"( )2H0
L2 !K2[ ]MI2 +
1
2Tr A
2( ) ! L2[ ] I I +1( ) !MI2[ ] ! 1
Kdet A( ) 2MS !1( )MI
# $ %
& ' (
And the probability for a transition (ΔMS = 1, ΔMI = 0)
!
P MS " MS #1( ) =$ 2H1
2
8Tr g2( ) #
1
g2
r e †g4
r e ( )
%
& '
(
) * S S +1( ) # MS
2 + MS[ ]
Below we list a MATLAB procedure (single2) to calculate the resonance fields and the probabilities for transitions in a system of arbitrary electron spin S and arbitrary nuclear spin I and with arbitrary orientation orientation (cosθ, φ). To simulate an EPR spectrum of a mono-crystal with an eigen-system of axis of the g-tensor oriented (cosθ, φ) versus the direction of the magnetic field, one has to convolute this diagram
!
H MS " MS #1, MI , S, I( ) ;P MS " MS #1( ){ } using some appropriate form function e.g. a Lorentz function. function [h,tp]=single2(u,phi) % %------------------------------------------------------- % Get resonance fields h and transition probabilities tp %------------------------------------------------------- % Input % ===== % u = cos ( is polar angle) % φ = azimuthal angle % Global % ====== % ns = 2S+1 % ni = 2I+1 % g = g-tensor (necessarily diagonal) % gg = g^2 % tg2 = trace(g^2) % a2 = a^2 (a = a-tensor, not necessarily diagonal) % a4 = a^4 % ta2 = trace(a^2) % da = det(a) % omega = Microwave Frequency in [s-1] % global g gg tg2 a2 a4 ta2 da ni ns omega a=sqrt(1-u*u);r=[a*cos(phi) a*sin(phi) u];r=(r.*g)'; % Zeeman hq=3.3361172e-11;beta=4.65987e-5; qg2=r'*r;gbeta=sqrt(qg2)*beta;hze=hq*omega/gbeta; % transition probability fp=tg2-r'*gg*r/qg2; % prepare hfs qa2=r'*a2*r/qg2;qa4=r'*a4*r/qg2/qa2; % loop a=ni*ni-1;c=ns*ns-1;sqa2=sqrt(qa2);
mi=0.5*(ni+1); for ki=1:ni mi=mi-1;b=mi*mi; ms=0.5*(ns+1); for ks=2:ns ms=ms-1; % transition probability if ki==ni, tp(ks)=fp*(0.25*c-ms*ms+ms); end % HFS tmp=0.5*((qa4-qa2)*b+.5*(ta2-qa4)*(0.25*a-b)+da*mi*(ms+ms-1)/sqa2); hhf=-sqa2*mi/gbeta - tmp/gbeta/gbeta/hze; % resonnance field h(ki,ks)= hze + hhf; end end To obtain the EPR spectrum of a polycrystalline sample a spatial integration over all single crystal is needed, i.e. over the surface of the unit sphere:
!
"1# cos$ #1 and
!
0 " # " 2$ This task is computer time expansive. Practice has shown that the best procedure is Gauss-Legendre quadrature {cf. C. Daul et al. dans Computer Phys. Commun.; 21, (1981), 385}. This is easily done using the MATLAB function gauleg (vide infra) followed by a convolution of the resulting diagram with an appropriate lineshape function (Lorentz ou Gauss, MATLAB function fls) and finally a numerical differentiation-smoothing (MATLAB function derv1) concludes the job. The MATLAB script listed below will carry out this task for you. %====================================================== % Simulation of EPR spectra of polycrsitalline samples %====================================================== clear all global g gg tg2 a2 a4 ta2 da ni ns omega %------------------------- % Begin of Input %------------------------- % 2*I+1 ni=2; % 2*S+1 ns=2; % diag(G-tensor) g=[3.0918 1.9616 1.9298]; % A-tensor in [cm-1] %a=[0.01489 0 0; % 0 0.00895 0; % 0 0 0.00208]; a=[0 0 0; 0 0 0; 0 0 0]; % Microwave Frequency in [s-1] omega=9000000000.0; % Peak to Peak Width in [Gauss] ppw=200; %------------------------- % End of Input %------------------------- % initialize a=a+1e-20*eye(3); gg=diag(g.^2);tg2=trace(gg); a2=a*a;a4=a2*a2;ta2=trace(a2);da=det(a); %------------------------------------------------------------ % Hypothesis: The axis of g- and A-tensor coincide
% hence, integration is performed on one octant! %------------------------------------------------------------ % get hlow, hup, npt, nu and nf [h,tp]=single2(0.,0.);hxn=min(min(h(:,2:ns)));hxx=max(max(h(:,2:ns))); [h,tp]=single2(0.,0.5*pi);hyn=min(min(h(:,2:ns)));hyx=max(max(h(:,2:ns))); [h,tp]=single2(1.,0.);hzn=min(min(h(:,2:ns)));hzx=max(max(h(:,2:ns))); nux=fix(max([abs(hzx-hxn)/ppw;abs(hxx-hzn)/ppw]))+1; nuy=fix(max([abs(hzx-hyn)/ppw;abs(hyx-hzn)/ppw]))+1; nu=max([nux;nuy]); nf=fix(max([abs(hxx-hyn)/ppw;abs(hyx-hxn)/ppw]))+1; % Adjust number of angular points nu=2*nu;nf=2*nf; % Spectral boundaries are extended by 10*ppw hlow=min([hxn;hyn;hzn])-10*ppw; hup=max([hxx;hyx;hzx])+10*ppw; % There are 10 points per line npt=10*fix((hup-hlow)/ppw)+1; dh=(hup-hlow)/(npt-1); % print nu, nf, npt fprintf(' *** Gauss-Legendre quadrature rule *** \n') fprintf(' ------------------------------------------\n') fprintf(' # polar quadrature points %3i \n',nu) fprintf(' # azimutal quadrature points %3i \n',nf) fprintf(' # spectral points %3i \n',npt) fprintf(' ------------------------------------------\n') % get angular grid (Gauss-Legendre) % u=cos(theta) , 0<u<1 [u,wu]=gauleg(0,1,nu); % phi , 0<phi<pi/2 [phi,wf]=gauleg(0,0.5*pi,nf); % % --- get spectrum --- % % init y=zeros(1,npt);tinit=clock; % loop over angular grid for i=1:nu fprintf(' %5.1f percent achieved \n',100*i/nu) for j=1:nf [h,tp]=single2(u(i),phi(j)); % 3*ppw is used as a cutoff for the line shape function hmn=min(min(h(:,2:ns)))-3*ppw;hmx=max(max(h(:,2:ns)))+3*ppw; imn=fix((hmn-hlow)/dh)-1;imx=fix((hmx-hlow)/dh)+1; for ipt=imn:imx hpt=hlow+dh*(ipt-1); for ks=2:ns for ki=1:ni y(ipt)=y(ipt)+wu(i)*wf(j)*tp(ks)*fls((hpt-h(ki,ks))/ppw); end end end end end telaps=etime(clock,tinit);minuts=fix(telaps/60);seconds=telaps-minuts; fprintf(' ------------------------------------------\n') fprintf(' Elapsed time: %6i min %4.1f sec \n',minuts,seconds) scale=max(abs(min(y)),max(y)); y=(1/scale)*y; % %plot([hlow:dh:hup],y),pause % take 1st derivative z=(1/dh)*derv1(y); plot([hlow:dh:hup],z)
function z=derv1(y) % Get 1st derivative of y % Mohos's method, integration uses Simpson's rule npt=length(y);mu=10;mu=2*fix(mu/2);n2=npt-mu; % a=2/(3*mu^4); for j=1:mu-1 if rem(j,2)==0, p(j)=2*a*j*j; end if rem(j,2)==1, p(j)=4*a*j*j; end end p(mu)=a*mu*mu; % z=zeros(1,npt); for i=mu+1:n2 for j=1:mu z(i)=z(i)+p(j)*(y(i+j)-y(i-j)); end end % z has to be divided by increment h=x(i+1)-x(i) function y=fls(x) % Lorentz %y=1./(1+4*x.*x); % Gauss y=exp(-2.773*x.*x); function [x,w]=gauleg(x1,x2,n) % get Gauss-Legendre Quadrature nodes and weights eps=3e-14;m=fix((n+1)/2);xm=0.5*(x2+x1);xl=0.5*(x2-x1); for i=1:m z=cos(pi*(i-0.25)/(n+0.5));z1=3; while abs(z-z1) > eps, p1=1;p2=0; for j=1:n p3=p2;p2=p1;p1=((2*j-1)*z*p2-(j-1)*p3)/j; end pp=n*(z*p1-p2)/(z*z-1);z1=z;z=z1-p1/pp; end x(i)=xm-xl*z;x(n+1-i)=xm+xl*z; w(i)=2*xl/((1-z*z)*pp*pp);w(n+1-i)=w(i); end To conclude this paragraph, the fig. below represents the simulation of the EPR spectrum of a polycristalline Co(acetylacetonate-ethylene-diamine) sample magnetically diluted in a host matrix ofde Ni(acetylacetonate-ethylene-diamine). The parameters of the simulation are those given in the MATLAB script above .
Fig. Simulation of the EPR spectrum of a polycristalline Co(acetylacetonate-ethylene-diamine) diluted magnetically in a host lattice of Ni(acetylacetonate-ethylene-diamine)
Expressions for magnetic coupling parameters 2nd order perturbation theory: Zeeman Splitting
!
gij = ge"ij + ge
#$k %k rk( )
) L i
k #µk &µ
) L j
k' #$k'
'$ ('µk'
)k
)µ *$( ))
Where: i, j = x, y, z ge = g-value of free electron = 2.00231931
!
rk =v r "
r R k , is the distance to the nucleus k
the
!
k" runs over all atoms (nuclei)
ν is the index of the singly occupied MO
!
"#k
= c$# %$$&k
' is that part of the ν-th MO whose basis functions χλ have their origin on nucleus k
!
"µ = c#µ$##
% , is the complete µ-th MO
!
"k rk( ) =1
rk
dV
drk
#$k1
rk
%
& '
(
) *
3
, is the radial factor of the spin-orbit coupling operator
!
) L i
k , is the i-th component of the angular momentum operator centred on atom k.
!
"µ is the µ-th Kohn-Sham eigenvalue
2nd order perturbation theory: Hyper-Fine Splitting a) dipolar contribution
!
Aijk = P
k "#k Fij
k
rk( )3"#
k +
2Pk "#
k ' Fijk
rk( )3"#
k
k$k'
% +
Pk "#
k' Fijk
rk( )3"#
k '
k '$k
% +
Pk
2 "#k &k rk( )
) L i
k "µk
k%{ } "µk
) L j
k
rk( )3"#
k
'# ('µµ$#% +
Pk
i'ipq "#k &k rk( )
) L p
k "µk
k%{ } "µk Fqj
k
rk( )3"#
k
'# ('µp,q
%µ$#%
Where: i, j = x, y, z
!
rk =v r "
r R k , is the distance to the nucleus k
the
!
k" runs over all atoms (nuclei)
ν is the index of the singly occupied MO.
!
"#k
= c$# %$$&k
' is that part of the ν-th MO whose basis functions χλ have their origin on nucleus k
!
"k rk( ) =1
rk
dV
drk
#$k1
rk
%
& '
(
) *
3
, is the radial factor of the spin-orbit coupling operator
!
) L i
k , is the i-th component of the angular momentum operator centred on atom k.
!
Fijk
=3riri " r
2#ij
r2
, is the dipolar operator
!
Pk =
µ04"
#e#ngegk and µ0 is the magnetic permeability in free space, βe is Bohr's magneton, βn is
the nuclear magneton, ge is the g-value of the free electron, gk is the g-value of the nucleus k (gk=µk/Ik)
!
"µ is the µ-th Kohn-Sham eigenvalue
!
" ipq (Levi-Civita symol) is an alternating tensor defined as:
!
" ipq=
0 if any two of i, p, q are equal
+1 if ipq is an even permutation of xyz
#1 if ipq is an odd permutation of xyz
$
% &
' &
b) Fermi contact term:
!
Acontact
k =8"
3Pk#
v r k =
v 0 ( )
2
2nd order perturbation theory: Zero Field Splitting
!
Dij ="#
k $k rk( )) L i
k "µk "µ
k ' $k' rk '( )) L j
k' "#k'
%# &%µk '
'k
'µ (#( )'
Where: i, j = x, y, z
!
rk =v r "
r R k , is the distance to the nucleus k
the
!
k" runs over all atoms (nuclei)
ν is the index of the singly occupied MO
!
"#k
= c$# %$$&k
' is that part of the ν-th MO whose basis functions χλ have their origin on nucleus k
!
"k rk( ) =1
rk
dV
drk
#$k1
rk
%
& '
(
) *
3
, is the radial factor of the spin-orbit coupling operator
!
) L i
k , is the i-th component of the angular momentum operator centred on atom k.
!
"µ is the µ-th Kohn-Sham eigenvalue
Quadrupolar Splitting: Electric Field Gradient (EFG)
!
PijK =
eQ
2I 2I "1( )VijK
e is the charge of an electron I is the nuclear spin Q is the nuclear quadrupole moment, defined as:
!
I,I QzzKI,I
and
!
VijK =
e
4"#0ZA
3 XiX j( )AK
$%ijRAK2
RAK5
A&K
' $
e
4"#00,m(
3 xi x j( )Kk$%ijrKk
2
rKk5
k
occ
' 0,m(
Where: i, j = x, y, z
!
r R AK
is the radius vector connecting the nuclei A and K
!
Xi( )AK
is its component along the molecular i-axis
!
r r Kk
is the radius vector connecting the nucleus K and the electron k and
!
xi( )Kk
is its i component ZA is the nuclear charge
!
0,m" is the ground state determinantal:
!
"1"2..."2n#1 Writing MO's as linear combination of AO's χk, we can split up the expression above in terms involving a different number of centres
!
VijKel( ) = "
e
4#$0Nµ ckµck 'µ %k
FijK
rK3%k'
k,k '
K
&' ( )
* ) µ
& +
2 caµckµ %aFijK
rK3%k
a
A
&A+K&
k
K
& +
caµca'µ %aFijK
rK3%a'
a,a'
A
&A+K& +
caµcbµ %aFijK
rK3%b
b
B
&a
A
&B+KB
&A+KA +
&
,
- )
. )
Where:
Nµ is the occupation of MO:
!
"µ = ckµ#kk
K
$K
$ , expanded in terms of AO's χk centred on atoms K.
2nd order perturbation theory: Remarks (i) Gauge invariance: 1st order + 2nd order Perturbation Theory! 1st order term is negligible (natural gauge) (ii) The sun over states includes all singly excited states i.e.: (unpaired electron)
"1# "1$ ..."k# "k$..."n%1$ "n&0 '"1# "1$ ..."k& k ..."n%1$ "n# "n$ (electron in doubly occupied MO)
!
" (empty MO)
!
"1# "1$ ..."k# "k$..."n%1$ "n&0 '"1# "1$ ..."k& k ..."n%1$ "n&0 "h&h (iii) It is important to remember that these excited state wave functions are not necessarily the best description of the physical excited states of the system. The only condition required by perturbation theory is that they form (or, in practice, approach) a complete set together with the ground state functions. This does correspond to uncoupled Hartree-Fock-Slater Theory. One-electron Spin-Orbit Matrix Elements (i) The r-3 behaviour of ξ(r( weighs the region close to the nucleus very heavily, hence only one-centre matrix elements need to be considered. (ii) The radial part of the spin orbit matrix elements are obtained by scaling r-3 matrix elements in the given set of basis functions in such a way that atomic spin-orbit splitting would be reproduced. I. e. the proportionality constant α in
!
" rs =Rr # Rs
Rr Rs
=$Rr r
%3Rs
Rr Rs
is the same as in
!
a " a =# a r$3a
Quasi Degenerate Ground-State: Kramers Doublet Consider the multiplet wave functions
!
i" = # $ m$ S mS
arising from a given configuration α are obtained by vector coupling as
!
i" = Aiµ #µµ
$
where: Γ is the label of the irreducible representation of the space part of the wave function mΓ refers to its component in case of degeneracy, S is the spin part of the wave function with component mS in case of spin multiplicity larger than 1
!
"µ = #1 #2 #3 ... is a single-determinant wave function of spin-orbitals χ1, χ2, χ3, ... Aiµ is an orthogonal square matrix of symmetry coefficients. The energies of this multiplets can be obtained to 1st order as:
!
E "i( ) = "i H0 "i = Fij E #µ j( )j=1
r
$
where H0 is classical hamiltonian without spin-orbit coupling. In order to include the spin-orbit coupling, we consider a perturbation
!
H1 = "k rk( )r L #
r S
k
$
Thus, we calculate:
!
"i H1 "j = AiµAiµ #µ H1 #$µ ,$
%
where:
!
"µ H1 "# =
$k h1 $kk
% if µ = #
$k h1 $ l if "µ and "# differ by 1 spinorbital $k and $ l( )0 if µ and # differ by more than 1 spinorbital
&
'
( (
)
( (
Since
!
H1 = "k rk( )r L #
r S
k
$ and
!
"k rk( )# rk$3, only one-centre terms
!
"µk#k rk( )
r l $
r s "% k
need to be
calculated. Finally, diagonalizing
!
Hij = E0"i( ) + "i H1 "j yields the spin-orbit states and hence the
Kramers doublet of the ground state.
Quasi Degenerate Ground-State: g- and A-Tensor Consider the Kramers doublet for the ground-state:
