Rela%vity and mul%reference concerns CHEM 430 Spring 2016 Rela%vity 2 History •  Galileo (1632)
–  Fundamental laws of physics are the same in all inertial frames
•  Einstein (1905)
–  Postulates
•  1) The principle of relativity-The laws by which states of
physical systems undergo change are not affected, whether
these changes of state be referred to the one or the other of
two systems in uniform translatory motion relative to each
other
•  2) The principle of invariant light speed-"... light is
always propagated in empty space with a definite velocity
[speed] c which is independent of the state of motion of the
emitting body." i.e. light in vacuum propagates with the
speed c (a fixed constant, independent of direction) in at least
one system of inertial coordinates (the "stationary system"),
regardless of the state of motion of the light source.
3
Transforma%ons •  Galilean transformation
–  x’=x-vt
! x ' $ ! 1 −v $! x $
#
&=#
&#
&
t
'
0
1
t
"
% "
%"
%
•  Lorentz transformation
–  Describes how two observers' varying measurements of space
and time can be converted into each other's frames of
reference
–  Reflects the surprising fact that observers moving at different
velocities may measure different distances, elapsed times,
and even different orderings of events !
x' $ ! 1
−v $! x $
#
&=γ#
" t ' % " −v / c
&#
&
1 %" t %
–  γ is a relative scaling factor
•  γà1 at the non-relativistic limit (velocities are small
1
−
compared to the speed of light)
2
" v %2
γ = $1− 2 '
# c &
4 Incorpora%ng rela%vity •  Klein-­‐Gordon equa%ons –  Energy of rela%vis%c free par%cle…but may be a complex root –  Neglects half spin nature of electron •  Dirac equa%on –  Posi%ve probability for first deriva%ve with respect to %me –  Wavefunc%on becomes a 4-­‐vector (4-­‐component, 1 electron wavefunc%on=spinor) " Ψ %
$ 1 '
$ Ψ2 '
Ψ =$
'
Ψ
3
$
'
$ Ψ4 '
#
&
H Ψ = EΨ
n
D
H = ∑ H ext
(i) + ∑Vij
i=1
i< j
D
H ext
(i) = α • pi c + β mc 2 − ∑
Vij =
1
rij
Zµ
ri − Rµ
5 Rela%vis%c effects-­‐In general n2
Z
mZ
vn =
En = − 2
mZ
n
2n
m0
m0
m=
=
2
v
Z2
1−
1− 2
c2
n c2
rn =
Rrel/Rnon-­‐rel •  Kinema%c aka scalar –  Direct •  Contrac%on of inner s and p orbitals •  αZ4 –  Indirect •  s,p contract=more nuclear shielding=d,f relax •  Spin-­‐orbit –  Causes spliZng of states Z(a.u.)
6 Scalar rela%vis%c effects •  Douglas-­‐Kroll-­‐Hess Theory –  Only one-­‐electron operators are modified to account for scalar effects (SOC requires two components) –  DKH Hamiltoniansàtransform one-­‐electron terms, minimal picture change effect –  There exist basis sets calibrated for these types of calcula%ons (e.g., cc-­‐pVTZ-­‐DK) H = ∑!"Ei +Veffsf (i)+ H SO (i)#$ + ∑!"Veffsf (i, j)+ H SO (i, j)#$
i
1 electron terms i≠j
2 electron terms 7 Spin-­‐orbit coupling-­‐Defini%on •  Interaction between the electron-spin and the orbital
angular momentum
•  Causes shifts in an electron's atomic energy levels due to
electromagnetic interaction between the electron's spin
and the nucleus's magnetic field
•  New (magnetic) interactions in the Hamiltonian operator
due to electron spin
•  SOC destroys the picture of an orbital having a definite
spin
•  No spin magnetic interactions=our Hamiltonian and total
electronic spin S2 commute
•  2s+1-fold degenerate with respect to the energy…spin
multiplet
•  SOC=relativistic effect, increases with Z
8 Stern Gerlach experiment •  1921, provided experimental basis for concept of electron spin
•  Set out to measure the magnetic moment of silver atoms by
deflecting a beam of silver atoms in an inhomogeneous
magnetic field
•  Expected one resulting smear along an axis
•  Saw two spots evenly separated on axis
•  àAg atoms have magnetic moment and it can adopt two
discrete values
9 Zeeman effect •  Modification of atomic or molecular spectrum by application of a
uniform magnetic field
•  Yields three lines where there is one in absence of magnetic field
•  Splitting pattern results from the interaction between the external
magnetic field and the orbital angular momentum of the atom
•  State with spatial angular momentum quantum number L splits
in 2L+1 components
∆ML=0,±1
10 Spin=Quantum effect •  Referred to as intrinsic angular momentum…BUT no
classical analog
•  Particles with difference spins (0,1/2,1,3/2) described by
different relativistic equations
•  Dirac equation= spin of ½
•  Definition for the spin of a particle with rest m≠0:
Spin is the angular momentum of a particle at rest.
11 Angular momentum Orbital Spin •  Observable (real-valued)
•  Vector notation:
→
∧
→
∧
→
∧
→
∧
→
∧
ℓ = r × p = −"(r × ∇)
•  Cannot determine all three
components simultaneously
–  Due to commutation of
operators
–  Results in a cone shape
∧ ∧2
∧2
•  Spin eigenfunc%on (|sms>) not a func%on of spa%al coordinates àspinor •  There exists Pauli spin matrices (introduced before) (
*
**
sms = )
*
*
*+
1
2
1
2
1
2
−
1
2
! 1 $
=#
&=α
" 0 %
! 1 $
=#
&= β
" 0 %
∧2
ℓ = ℓ x + ℓ y + ℓz
12 Spin Hamiltonian •  SOC arises in Dirac theory •  s and l not separately conservedàuse j 1
H SO =
2mec 2
∧
1 # ∂V & ∧ ∧
% ( L• S
r $ ∂r '
o  V=Coulomb poten%al of the electron in the field of the atom o  SOC propor%onal to L·∙S ∧ ∧
∧
∧
∧
&
1#
L• S = % J − L − S (
'
2$
o  With spin-­‐orbit operator, matrix elements between singlet and triplet may not be zero 3
∧
Σ H SO 1 Σ+0 ≠ 0
−
0
13 Spin Hamiltonian-­‐Visual 14 Spin-­‐spin coupling •  Involves only valence orbitals
•  Contributions of closed-shell core orbitals cancel
•  Therefore, remains nearly constant with increasing atomic
number
•  Difficult to distinguish (experimentally) between second
order SOC and first order SSC
15 Russell Saunders overview RS term symbol (2S+1)L •  assume that spin-­‐spin coupling > orbit-­‐orbit coupling > spin-­‐orbit coupling •  Spin-­‐Spin coupling S -­‐ spin quantum number, summa%on of individual ms values, result of coupling of spin quantum numbers for the separate electrons. •  Orbit-­‐Orbit coupling L -­‐ the total orbital angular momentum quantum number, defines the energy state for a system of electrons •  Spin-­‐Orbit coupling Coupling of L and Sàresults in J, Mul%plicity (2S+1) occurs when several levels are close together 16 Terms arising from configura%ons Configuration
S
L
Term
s1
½
0
2S
s2
0
0
1S
s1s1
0 or 1
0
1S, 3S
s1p1
0 or 1
1
1P, 3P
p 1p 1
0 or 1
0, 1, or 2
1D, 1P, 1S
p2
0 or 1
0, 1, or 2
1S, 3P, 1D
p3
?
?
?
3D, 3P, 3S
17 Russell-­‐Saunders examples •  N-atom configuration
•  Ground state
1s22s22p3
•  Terms…
4S, 2D, 2P o  1st excited state 1s22s22p23s1
o  Terms…
4P, 2P o  Hund’s rules
o  Ground term with max spin multiplicity=lowest energy
o  If same spin multiplicity, then largest L value
18 SOC pertaining to RS •  Total angular momentum (J) is added as a subscript to term
symbol
•  |L-S| ≤ J ≤ L+S
•  Example: 2P
–  Breaks into 2P1/2 and 2P3/2
19 Spectroscopy •  Spin forbidden transitions may yield some intensity due to
magnetic interactions coupling states of different multiplicities
•  More prevalent further down periodic table (due to increase in
SOC)
•  Spin-allowed=internal conversion (IC)
•  Spin-forbidden=intersystem crossing (ISC)
Jablonski diagram 20 WHERE RS FAILS…
•  RS is a good approximation for first row TMàspin-orbit (J)
coupling can generally be ignored
•  However …Z>30àSOC becomes more significant and the jj coupling scheme is used
21 jj-­‐coupling •  Lower case j indicates angular momentum of one electron
(j=l+1/2)
•  J1=L1+S1
J2=L2+S2 …
J=ΣiJi
•  S1p1
–  S electronàl1=0, s1=1/2
–  P electronàl2=1, s2=1/2
–  Possible ji valuesàli+si, li+si-1, …|li-si|
–  For first electron, j1=1/2
–  For second electron, j2=3/2 and j2=1/2
–  Therefore, J values are: j1=1/2, j2=3/2àJ=2 and J=1
j1=1/2 j2=1/2àJ=1 and J=0
22 Atomic energy spectra •  Atomic energy levelsàdetermined by Charlotte Moore
•  Report all configurations and subsequent splitting of each
level
•  Can use tables to determine SOC of element
•  NIST atomic spectra database
–  Lines and levels available
–  Can see spectra for elements and include term energy
and J level
23 Atomic energy spectra Fe(I) 24 Mu%reference Mul%configura%on 25 MCSCF METHODS •  Single configura%on methods can fail for –  Species with significant diradical character –  Bond breaking processes –  Open for excited electronic states –  Unsaturated transi%on metal complexes àThen MCSCF method is necessary •  Most common approach •  Complete ac%ve space SCF (CASSCF/FORS) •  Ac%ve space = orbitals+electrons involved in process •  Full CI within ac%ve space: op%mize orbitals & CI coeffs •  Size-­‐consistent 26 Mul%reference methods •  Mul%reference methods are needed for: –
–
–
–
Near-­‐degeneracy Bond breaking Excited states radicals •  Nondynamical correla%on –  MCSCF •  Dynamical correla%on –  Varia%onal: MRCI –  Based on perturba%on theory: CASPT2 , MS-­‐CASPT2, MRMP2 –  Not widely spread yet: MRCC, MRCI/DFT 27 Correla%on •  Dynamical –  Electrons instantaneously avoiding each other •  Non-­‐dynamical –  Nearly degenerate electron configura%ons –  Include all nearly degenerate electron configura%ons in star%ng (reference) wavefunc%on –  Find orbitals that minimize mixture of near-­‐degenerate determinants (MCSCF) In prac%ce… (1)  U%lize MCSCF for non-­‐dynamical correla%on (2)  Employ correlated methods (MRCI, MRCC…) for dynamical correla%on 28 Mul%configura%on Self-­‐Consistent Field Theory (MCSCF) •  Construct spin adapted linear combina%on of Slater determinants CSFs
ΨMCSCF = ∑ c n CSF
n=1
€
•  Two op%miza%ons have to be performed –  Op%mize the MO coefficients –  Op%mize the expansion coefficients of the CSFs 29 Orbital spaces CASSCF Virtual orbitals CAS Double occupied orbitals 30 Choosing the ac%ve space Ac%ve space selec%on will dictate the accuracy of the method Must u%lize your knowledge of the system to define ac%ve space Small systemsàall valence orbitals can be included in the ac%ve space Conjugated systemsàall π orbitals if possible should be included in the ac%ve space •  Heteroatomic ringà the lone pairs should be included also. Choose the ac%ve orbitals •
•
•
•
Depends on the problem and the ques%ons being asked For a π system all π orbitals should be included if possible If bond breaking include bonding, an%bonding Check occupa%on numbers of orbitals (between 0.02 -­‐ 0.98) Trial and error n!(n + 1)!
N=
" m% " m % "
(m,n): distribute m electrons in n orbitals m% "
m %
!
+
1
!
n
−
!
n
−
+ 1'!
$ '$
'$
'$
# 2&# 2 &#
&
#
2
2 &
31 State averaged MCSCF •  All states of interest must be included in the average •  When the poten%al energy surface is calculated, all states of interest across the coordinate space must be included in the average •  State-­‐averaged MOs describe a par%cular state poorer than state-­‐specific MOs op%mized for that state •  State-­‐average is needed in order to calculate all states with similar accuracy using a common set of orbitals. This is the only choice for near degenerate states, avoided crossings, conical intersec%ons. •  Provides common set of orbitals for transi%on dipoles and oscillator strengths 32 Mul%-­‐reference configura%on interac%on (MRCI) •  A reference space is needed similar to the ac%ve space at MCSCF •  References are created within that space Frozen Virtual orbitals
Virtual orbitals CAS •  Single and double excita%ons using each one of these references as a star%ng point •  ΨMRCI=∑ciΨI •  Includes dynamical correla%on beyond the MCSCF •  Orbitals from an MCSCF are used for the subsequent MRCI •  The states must be described qualita%vely correct at the MCSCF level DOCC orbitals Frozen orbitals 33 CASPT2 •  Second order perturba%on theory is used to include dynamic correla%on •  Widely used for medium size conjugated organic systems •  Errors for excita%on energies ~0.3 eV •  There are no analy%c gradients available so it is difficult to be used for geometry op%miza%ons and dynamics 34