Relaxation theory of the electronic spin of a complexed paramagnetic ...

P. H. Fries and E. Belorizky

J. Chem. Phys. (2007)

Supplementary Material S1

Relaxation theory of the electronic spin of a complexed paramagnetic metal ion in solution beyond the Redfield limit Pascal H. Fries1 Laboratoire de Reconnaissance Ionique et Chimie de Coordination, Service de Chimie Inorganique et Biologique (UMR-E 3 CEA-UJF), CEA/DSM/Département de Recherche Fondamentale sur la Matière condensée, CEA-Grenoble, F-38054 Grenoble Cédex 9, France

Elie Belorizky Laboratoire de Spectrométrie Physique, CNRS-UMR 5588, Université Joseph Fourier, BP 87, F-38402 SaintMartin d’Hères Cédex, France

Brief reading guide The main results of the Supplementary Material are given in Appendices A and B. Appendix A contains a general proof beyond the Redfield limit of the symmetry properties Eq. (16) of the time correlation functions (TCFs). The proof follows the employed simulation procedure in the limit of an infinite number of Brownian trajectories. In Appendix B the time-evolution of the populations of the eigenstates of the Zeeman Hamiltonian H 0 is described by a new theory accounting for the time-modulation of the Zeeman transition frequencies by the terms of the fluctuating zero-field splitting (ZFS) Hamiltonian which commute with H 0 . The supplementary Appendices SA, SB, SC, and SD can be skipped in a first reading. Appendix SA gives a proof of additional symmetry properties of the TCFs at zero field. Appendix SB reports on mathematical techniques used to derive the analytical expressions of the population evolution discussed in Appendix B. The influence of the rhombicity of the fluctuating ZFS Hamiltonian is also studied in Appendix SB. Appendix SC is a reminder of the equivalence between the Redfield theory and the usual population evolution formalism within the Redfield limit. Appendix SD gives the Redfield relaxation matrices needed to calculate the Redfield TCFs which are compared with the simulation results in the printed article. The longitudinal relaxation is also studied when the static ZFS Hamiltonian is rhombic.

1

Electronic mail : [email protected]




APPENDIX A: ON THE SYMMETRY PROPERTIES OF THE g kkqq'' (t )

Consider the symmetry properties of the statistical operator σ (t ) defined by Eq. (6) under the rotation group G of the whole statistical ensemble of N sys realizations of the system. In an

external magnetic field B 0 , G is the rotation group C∞ about its direction z. When B 0 vanishes, G is the full rotation group SO(3). Under a geometric rotation R of G, the density matrix σ is transformed into Rˆ (σ ) = Rσ R † , where R is the associated quantum rotation matrix.1 We will prove that the transformation of σ (t ) under Rˆ at time t is simply given by the evolution of the transformation of σ (0) under the same rotation, namely the following commutation property holds ˆ [σ (0)] = ERˆ [σ (0) ] RE

(A1)

By definition, the statistical operator ERˆ [σ (0) ] is

1 ERˆ [σ (0) ] ≡ N sys

∑U

j

(t )Rσ (0)R †U j (t )†

(A2)

j

Whatever the B 0 value, the ensemble of rotational trajectories trajec ( j ) of the various realizations j of the metal complex has an isotropic character. Thus, each trajectory trajec ( j ) has to be the image of another trajectory trajec ( i ) under the geometric rotation R , i.e. trajec ( j ) = R ⎡⎣ trajec ( i ) ⎤⎦ . Then, at each time t, the ZFS Hamiltonian of the realization j in the (L) (L) laboratory frame H ZFS, j (t ) is also the image of the Hamiltonian H ZFS,i (t ) under R , i.e.

(L) (L) † H ZFS, j (t ) = RH ZFS,i (t ) R

(A3)




In the presence of the magnetic field B 0 , the Zeeman contribution H 0 to the total Hamiltonian is only invariant under the rotation subgroup C∞ , so that the total Hamiltonian in the laboratory frame (L) H j (t ) = H 0 + H ZFS, j (t )

(A4)

(L) is the image of its counterpart H i (t ) ≡ H 0 + H ZFS, i (t )

H j (t ) = RH i (t )R †

(A5)

under the sole rotations R of C∞ . Now, the evolution operator U j (t ) induced by H j (t ) is simply the transformation under the rotation R of the evolution operator U i (t ) induced by H i (t ) , i.e. U j (t ) = RU i (t )R †

(A6)

From Eq. (A6) and the definition (6) of σ (t ) , the statistical operator of Eq. (A2) can be rewritten as

1 ERˆ [σ (0) ] = N sys

∑ ⎡⎣ RU (t )R i

i

†

⎤⎦ ⎡⎣ Rσ (0)R † ⎤⎦ ⎡⎣ RU i (t )R † ⎤⎦

†

(A7)

ˆ [σ (0) ] ≡ Rσ (t )R = RE †

Property (A1) is proven in the presence of B 0 . It can be readily extended to the full proper rotation group SO(3) in the absence of magnetic field, since Eq. (A7) holds for any rotation

R of this group.

kk ' Turn to the symmetry properties of g qq ' (t ) . Let R be any rotation leaving the spin

system and the magnetic field invariant. Eq. (A1) can be rewritten as



⎡ 1 ⎣⎢ N sys

σ (t ) ≡ E [σ (0)] = Rˆ −1 ERˆ [σ (0) ] = R † ⎢


⎤ † † U t R σ R U t ( ) (0) ( ) ⎥R ∑i i i ⎦⎥

(A8)

The coefficients σ kq (t ) of σ (t ) read

{

}

†

σ kq (t ) ≡ tr ⎡⎣tkq ⎤⎦ σ (t )

⎧⎪ ⎡ 1 † = tr ⎨ ⎡⎣tkq ⎤⎦ R † ⎢ ⎢⎣ N sys ⎩⎪

⎤ ⎫⎪

∑U (t )Rσ (0)R U (t ) ⎥ R ⎬ †

†

i

i

i

⎧⎪ † ⎡ 1 = tr ⎨ ⎡⎣ Rtkq R † ⎤⎦ ⎢ ⎢⎣ N sys ⎪⎩

⎥⎦ ⎭⎪

(A9)

⎤⎫ ⎛ † † ⎪ q' q' ⎞ ( ) (0) ( ) R R U t σ t U t ⎥⎬ ∑i i ⎜ ∑ k' k' ⎟ i ⎥⎦ ⎪⎭ ⎝ k ', q ' ⎠

Now, the tensor operators tkq and tkq'' transform under the rotation R as

Rtkq R † = ∑ Dqk1q ( R )tkq1 and Rtkq'' R † = ∑ Dqk'1' q ' ( R )tkq''1 q1

(A10)

q '1

The coefficients σ kq (t ) become

σ kq (t ) = ∑ σ kq'' (0) ∑ Dqk q ( R)* Dqk'' q ' ( R) g kkq q' ' (t ) 1 1

1

k ', q '

1

(A11)

q1 , q '1

Since the expressions (A11) and (11) of σ kq (t ) are equal for any initial values σ kq (0) of the statistical operator, the TCFs have to satisfy the equalities g kkqq'' (t ) =

∑D

k q1q

( R )* Dqk'1' q ' ( R ) g kkq1q' '1 (t )

(A12)

q1 , q '1

Whatever the field magnitude B0 , each coefficient σ kq (t ) has to keep the same value under the rotations R of the subgroup C∞ . Assume that R is a rotation of angle α about the z axis. The TCFs g kkqq'' (t ) should verify the equations


J. Chem. Phys. (2007) g kkqq'' (t ) = exp [i (q − q ')α ] g kkqq'' (t )

Supplementary Material S5 (A13)

which hold for all values of α , so that the indexes q and q' of the TCFs different from zero must be equal. Thus, for B 0 ≠ 0 , the general expression (11) of the coefficients σ kq (t ) in terms of the TCFs reduces to

σ kq (t ) = ∑ σ kq' (0) g kkqq' (t ) k'

(A14)




APPENDIX SA: SYMMETRY PROPERTIES OF THE g kkqq'' (t ) AT ZERO FIELD

When B 0 = 0 , the value (A12) of g kkqq'' (t ) is independent of the rotation R of the proper rotation group SO(3). Then, it is equal to the average of the values corresponding to all the rotations of SO(3), i.e. ⎡ 1 g kkqq' (t ) = ∑ ⎢ 2 q1 ⎣ 8π =

δ kk '

⎤ Dqk1q ( R )* Dqk1'q ( R)d Ω R ⎥ g kkq1q' 1 (t ) SO(3) ⎦

∫

∑

2k + 1 − k ≤ q1 ≤ k

g

q1q1 kk

(SA1)

(t )

From Eq. (SA1) the TCFs different from zero must have equal indexes k and k'. Furthermore, the TCFs g kkqq (t ) are equal for different values of the index q. Thus, for B 0 = 0 , the general expression (11) of the coefficients σ kq (t ) in terms of the TCFs reduces to

σ kq (t ) = σ kq (0) g kk00 (t )

(SA2)




APPENDIX B: POPULATION EVOLUTION AND TRANSITION PROBABILITIES

Let M and EM = M =ω0 be the eigenstates and energies of the Zeeman Hamiltonian =H 0 of a spin S. Consider a general stationary random time-dependent Hamiltonian =H1(L) (t ) acting on S, of zero mean value =H1(L) (t ) = 0 , and correlation time τ c . When H1(L) (t ) τ c is sufficiently weak, it is well-known2-4 that =H1(L) (t ) induces a transition probability per unit time WMM 0 from any state M 0 to a different state M and that these transition probabilities govern the time-evolution of the populations of the states M . The aim of this appendix is to reconsider the physical assumptions and mathematical properties underlying the concept of transition probabilities in order to show how it applies at sufficiently high field even if H1(L) (t ) τ c is not weak. A careful presentation of the theory of the evolution of the populations of the eigenstates M of the Zeeman Hamiltonian H 0 is needed for a rigorous derivation of this property. In particular, the elementary time-dependent perturbation theory has to be revisited.

The evolution of the populations of the eigenstates M of H 0 caused by the random perturbing Hamiltonian H1(L) (t ) can be conveniently expressed within the framework of the density matrix theory. At a given time ta ≥ 0 assume that the populations of the subensembles of realizations j of the spin system are represented by the same diagonal part of the density matrix

σ diag (ta ) = ∑ σ M M (ta ) M 0 M 0 0

(B1)

0

M0

Under the combined actions of the Zeeman Hamiltonian H 0 and perturbing Hamiltonian H1(L)j (t ) each normalized state M 0

evolves at a later time tb ≥ ta to a normalized state

ψ j , M (tb ) such as ψ j , M (ta ) = M 0 . Let c j , MM (tb , ta ) be the coefficients of ψ j , M (tb ) in 0

0

0

0

the basis M . As shown in the discussion of Property PA, the diagonal part of the density




matrix representing the ensemble of sub-ensembles j evolves independently of the offdiagonal terms of this matrix so that

σ diag (tb ) = ∑ σ M M (ta ) ψ j , M (tb ) ψ j , M (tb ) 0

0

0

0

M0

= ∑ σ M 0 M 0 (ta )∑ c j , MM 0 (tb , ta ) M 2

M0

M

(B2)

M

= ∑∑ c j , MM 0 (tb , ta ) σ M 0 M 0 (ta ) M M 2

M

M0

The populations σ MM (tb ) in the ensemble of sub-ensembles are given by

σ MM (tb ) = ∑ c j , MM (tb , ta ) σ M M (ta ) 2

0

0

(B3)

0

M0

Since H1(L) (t ) is stationary, the mean values c j , MM 0 (tb , ta )

2

depend only on the time

difference tb − ta . Because of the normalization of the states ψ j , M 0 (tb ) probabilities PMM 0 (tb − ta ) ≡ c j , MM 0 (tb , ta )

2

they represent

which are the populations of the states M in the

ensemble of sub-ensembles. Thus, Eq. (B3), which gives the evolution of the populations

σ MM , can be rewritten as σ MM (tb ) = ∑ PMM (tb − ta )σ M M (ta ) 0

0

0

(B4)

M0

(

)

Introducing the relaxation matrix P(t ) ≡ PMM 0 (t ) , Eq. (B4) can be rewritten in matrix form as

σ diag (tb ) = P(tb − ta )σ diag (ta )

(B5)

When H1(L) (t ) τ c is sufficiently weak, accurate second-order expansions in H1(L) (t ) of all the probabilities

PMM 0 (tb − ta ) are readily obtained2-4 for M ≠ M 0 from the first-order

approximations in H1,(L)j (t ) of the coefficients c j , MM 0 (tb , ta ) of ψ j , M 0 (tb ) and for M = M 0



from the exact relation PM 0 M 0 (tb − ta ) = 1 −

∑

M ≠M0


PMM 0 (tb − ta ) . This standard theory will now be

extended to arbitrary values of H1(L) (t ) τ c when the field is high enough. Since H1(L) (t ) is stationary, the theory is independent of the time origin and it is possible to set ta = 0 , tb = t for simplifying the notation. Assume that the spin state ψ j (t ) = U j (t ) M 0 originates from the eigenstate M 0 at time t = 0 . Its value at time t is obtained as an approximate solution of the Schrödinger equation i

d ψ j = ⎡⎣ H 0 + H1,(L)j (t ) ⎤⎦ ψ j dt

by expanding ψ j (t ) in the basis of

the eigenstates M of H 0 as3-5

ψ j (t ) ≡ ∑ exp ( −iM ω0t ) c j , M (t ) M

(B6)

M

Let ωMM 0 ≡ ( M − M 0 )ω0 be the transition frequency from the state M 0 to the state M . At time t = 0 , c j , M (t ) = δ MM 0 . As long as the weaker condition c j , M (t ) ≅ δ MM 0 derived by continuity holds, the coefficients c j , M (t ) for M ≠ M 0 can be approximated as

t

(

)

c j , M (t ) ≅ −i ∫ exp iωMM 0 t ' M H1,(L)j (t ') M 0 dt ' 0

(B7)

In the standard derivation of the transition probabilities all possible field values are treated globally.4 Then, the perturbing Hamiltonian H1,(L)j (t ) has to be weak so that c j , M (t ) ≅ δ MM 0 holds for a sufficiently long duration [ 0,tmax ] . Here, the considered field values are above a threshold B0min which will be determined now. As the magnetic field B0 increases, the key point is that ωMM 0 = M − M 0 ω0 also increases, giving rise to more and more rapid

(

)

oscillations of exp iωMM 0 t ' , which average out the integral (B7), i.e.,

c j , M (t ) → 0 as

B0 → ∞ . Thus, for any given value of tmax , there is a field threshold B0min , depending on the magnitude of H1(L) (t ) , such as the first-order approximation c j , M (t ) ≅ δ MM 0 is valid during the interval [ 0,tmax ] for B0 ≥ B0min . In what follows, for Eqs. (B8) to (B14), it is assumed that




either H1(L) (t ) is weak or B0 ≥ B0min , so that c j , M (t ) ≅ δ MM 0 holds during the interval [ 0,tmax ] . Then, if the spin of a given realization of the metal complex is in the Zeeman state M 0 at 2

with M ≠ M 0 at time t ≤ tmax is c j , M (t ) with

time 0, its probability to be in the state M

c j , M (t ) given by the approximate Eq. (B7). More generally, if the N sys realizations j of the metal complex characterized by perturbing Hamiltonians H1,j(L) (t ) have their spins in the Zeeman state M 0 at time 0, the probability PMM 0 (t ) to have their spins in the state M with M ≠ M 0 at time t ≤ tmax is

2

PMM 0 (t ) ≡ c j , M (t ) ≡

1 N sys

∑ c j ,M (t ) ≅ 2

j

∫

t 0

(

)

exp iωMM 0 t ' M H1,(L)j (t ') M 0 dt '

2

(B8)

where the bar is the average over the N sys realizations.

The probability PMM 0 (t ) can be expressed in terms of time correlation functions of H1(L) (t ) . Following the usual procedure this is done by using the derivative of PMM 0 (t ) with respect to time. The time derivative of PMM 0 (t ) is

∗

t

dPMM 0 (t ) dt ≅ 2 Re ∫ exp(iωMM 0τ ) M H1,(L)j (t ) M 0 M H1,(L)j (t − τ ) M 0 dτ 0

(B9)

Introduce the stationary TCF

kMM 0 (τ ) ≡ M H1,(L)j (t ) M 0 M H1,(L)j (t − τ ) M 0

∗

(B10)

of the random variable M H1(L) (t ) M 0 . The TCF kMM 0 (τ ) has a correlation time τ c such as kMM 0 (τ ) ≅ 0 for τ >> τ c . As it is the case for many physical Hamiltonians H1(L) (t ) , it is assumed that kMM 0 (τ ) is a decaying monoexponential k MM 0 (τ ) = k MM 0 (0) exp ( − t τ c ) so that Eq. (B9) becomes




dPMM 0 (t ) dt ≅ 2kMM 0 (0) Re ∫ exp(iωMM 0τ − τ τ c )dτ t

(B11)

0

Define the spectral density wMM 0 (ω ) of kMM 0 (τ ) as

∞

wMM 0 (ω ) ≡ 2 ∫ cos(ωτ ) kMM 0 (τ ) dτ = 2k MM 0 (0) 0

τc 1+ω 2τ c2

(B12)

The expression of the derivative dPMM 0 (t ) dt in terms of wMM 0 (ω ) , which is obtained by direct integration of Eq. (B9), can be replaced in the equation PMM 0 (t ) =

∫

t 0

⎡⎣ dPMM 0 (t ') dt '⎤⎦ dt ' to

yield the formula ⎡ ζ (t ,τ c , ωMM 0 ) ⎤ PMM 0 (t ) = wMM 0 (ωMM 0 ) ⎡⎣t − ζ (t ,τ c , ωMM 0 )τ c ⎤⎦ = wMM 0 (ωMM 0 )t ⎢1 − ⎥ t τc ⎣ ⎦ with

ζ (t ,τ c , ω ) ≡

{

(B13)

}

1 1 − ωˆ 2 − exp ( − t τ c ) ⎡⎣(1 − ωˆ 2 ) cos(ωt ) − 2ωˆ sin(ωt ) ⎤⎦ (ωˆ = ωτ c ) 2 1+ωˆ

The correction function ζ (t ,τ c , ω ) has a complicated time-dependence, but ranges in the interval [ −2,1] . Its series expansion about t = 0 is ζ (t ,τ c , ω ) = ( t τ c ) − (1 2 ) (1 + ωˆ 2 ) ( t τ c ) . 2

Furthermore, the approximate expression of PMM 0 (t )

2 ⎡ 1 − ωMM τ2 1 ⎤ 0 c PMM 0 (t ) ≅ wMM 0 (ωMM 0 ) t ⎢1 − ⎥ for t ≥ 5τ c 2 2 ⎢⎣ 1 + ωMM 0τ c t τ c ⎥⎦

(B14)

has an accuracy better than 1%. Then, we have the important property: For M ≠ M 0 , the probability PMM 0 (t ) ≅ wMM 0 (ωMM 0 )t is simply proportional to time t to within a relative error of

ε for t ≥ max (1 ε ,5 )τ c , so that wMM (ωMM ) represents a transition probability per unit time 0

0

(transition rate) to within this precision. As discussed above, Eq. (B13), which is equivalent to Eq. (B8), is valid provided that cM (t ) ≅ δ MM 0 holds during the interval [ 0,tmax ] . For the N sys realizations j of the metal complex this condition transforms for M ≠ M 0 into PMM 0 (tmax ) ≅

P. H. Fries and E. Belorizky wMM 0 (ωMM 0 )tmax > τ 2 . The comparison




is carried out for a spin S = 3 2 , a magnetic field B0 = 7.05 T, and the fluctuating ZFS (L) (t ) of Sec. III.B. This perturbing Hamiltonian gives rise to two different Hamiltonian H ZFS

approximate (a) probabilities P1a (t ) ≡ P−a1 2,−3 2 (t ) and P2a (t ) ≡ P1a2,−3 2 (t ) (a = long-time, standard, mod) of associated transition probabilities w1 ≡ w−1 2,−3 2 (ω0 ) = ( 6 5 ) j (ω0 ) and w2 ≡ w1 2,−3 2 (2ω0 ) = ( 6 5 ) j (2ω0 ) , j (ω ) being defined by Eq. (27). The time-dependent relaxation

(

)

matrix P ≡ PMM 0 governing the population evolution is

⎛1 − P1a − P2a ⎞ P1a P2a 0 ⎜ ⎟ P1a 1 − P1a − P2a 0 P2a ⎜ ⎟ P= ⎜ ⎟ P2a 0 1 − P1a − P2a P1a ⎜⎜ ⎟ 0 P2a P1a 1 − P1a − P2a ⎟⎠ ⎝

(B21)

(

Besides, the usual time-independent Redfield relaxation matrix W ≡ wMM 0

)

of the

populations is according to Eq. (B19) ⎛ − w1 − w2 ⎜ w1 W≡⎜ ⎜ w2 ⎜ 0 ⎝

w1

w2

− w1 − w2

0

0 w2

− w1 − w2 w1

⎞ ⎟ w2 ⎟ w1 ⎟ ⎟ − w1 − w2 ⎠ 0

(B22)

The probabilities P1a (t ) and P2a (t ) derived from the various theoretical approximations are compared at short time t ≤ τ 2 4 = 25 ps in Fig. S1. Both Pqstandard (t ) ( q = 1, 2) given by Eq. (B13) and Pqmod (t ) given by Eqs. (B18) to (B20) rise steeply from zero during the very short durations π ( qω0 ) and then oscillate around their long-time limits wq (t + τ 2 ) which are derived from Eq. (B14) and exceed Pqlong-time (t ) by the quantity wqτ 2 of strong relative influence for t ≤ τ 2 . Moreover, because of the modulation of the transition frequencies ωMN (L) (t ) , the two probabilities Pqmod (t ) reach their long-time limits to within a few due to H ZFS

percents over durations < τ 2 , which are much shorter than those of the standard functions Pqstandard (t ) of the order of 10τ 2 according to Eq. (B14).




FIG. S1. S = 3/2, B0 = 7.05 T. Time variation of the approximate (a) probabilities P1a (t ) ≡ P−a1 2,−3 2 (t )

and

P2a (t ) ≡ P1a2,−3 2 (t )

(a = long-time, standard, ZFS-

modulated) defining the evolution of the TCFs g kk00' (t ) for a static axial ZFS Hamiltonian of parameters ∆S = 2 3DS = 1 cm-1, ES = 0 , fluctuating with a rotational correlation time τ 2 = 100 ps. In particular, the probability P1mod (t ) is well approximated by w1 (t + τ 2 ) for t ≥ tmin = 4π ω0 , where tmin < τ 2 is just the duration of two oscillation periods at the Larmor frequency ω0 ! Now, because w1 and w2 decrease with field as 1 (ω0τ 2 ) and 1 ( 2ω0τ 2 ) , respectively, the 2

2

relation P2a (t ) ≅ P1a (t ) 4 holds at high field as shown by the different scales of the vertical axes of Fig. S1. The relaxation matrix P is dominated by the terms P1a (t ) . Briefly, for tmin ≤ t ≤ tmax , P is of the form P ≅ 1 + W × (t + τ 2 ) to within a good approximation. Now, according to Eq. (B5), the evolution of the population vector σ diag (t ) is given by

σ diag (t ) = Pσ diag (0) . Then, the time variation of σ diag obeys (i) the usual Redfield equation

(

mod dσ diag dt = Wσ diag for t ≥ tmin and (ii) σ diag (t ) = P modσ diag (0) with P mod ≡ PMM 0

)

expressed

by Eq. (B21) for t < tmin . The evolution of σ diag (t ) given by

σ diag (t ) = exp ⎡⎣ P mod − 1⎤⎦ σ diag (0)

(B23)




satisfies both time conditions to second order in P1mod (tmin ) and P2mod (tmin ) . Since tmin < τ 2 , the approximation Pqmod (tmin ) ≅ wqτ 2 is valid and Eq. (B23) is accurate to within a relative error of about max ( wqτ 2 ) = ( w1τ 2 ) . According to Eq. (B21) the matrix P mod − 1 has the same 2

2

q =1,2

structure as the Redfield relaxation matrix W given by Eq. (B22). It also has the same eigenvectors given in Appendix SC, so that its exponential can be calculated in a similar way. Therefore, the "longitudinal" TCFs g kk00,mod (t ) derived from the evolution Eq. (B23) of σ diag are simply the Redfield expressions (34) to (36) with 2 P1mod (t ) replacing 1 T1e(1) = 2w1 and 2 P2mod (t ) replacing 1 T1e(2) = 2 w2 .

FIG. S2. S = 3/2, B0 = 7.05 T. Short-time comparison of the Monte-Carlo (MC) simulated TCFs g kk00 (t ) with their counterparts g kk00,mod (t ) derived from the evolution Eq. (B23) of the theory of ZFS-modulated transition frequencies for a static ZFS Hamiltonian of parameters ∆S = 2 3DS = 1 cm-1, ES = 0 , fluctuating with a rotational correlation time τ 2 = 100 ps.

The TCFs g kk00,mod (t ) displayed in Fig. S2 have the same damped oscillations as their simulated counterparts for t ≤ tmin and tend to the long-time limits wq (t + τ 2 ) in a very similar way. This clearly validates the present theory of ZFS-modulated transition frequencies. Furthermore, in Fig. S3 of Appendix SB it is shown that for a given magnitude ∆S of the




static ZFS the simulated TCFs g kk00 (t ) keep very similar time behaviors irrespective of the presence of the largest possible rhombic terms ES = DS 3 .

Practical analytical high-field expressions of the TCFs g kk00 (t ) are easily derived by neglecting their transient damped oscillations. For that purpose, the approximation

W × (t + τ 2 ) of P mod − 1 , which was shown to be valid for t ≥ tmin is extended to the whole time domain t ≥ 0 so that Eq. (B23) simplifies to

σ diag (t ) = exp [ W × (t + τ 2 ) ]σ diag (0)

(B24)

The expressions of the "longitudinal" TCFs derived from Eq. (B24) are simply given by their Redfield counterparts with the time argument t replaced by t + τ 2 . As discussed after Eq. (B23), the relative error of these TCFs is about ( w1τ 2 ) . The same relative error is introduced 2

by the fundamental Eq. (B7) of the elementary time-dependent perturbation theory which rests on the assumption c j , M 0 (t ) = 1 and which thus amounts to neglecting the mean correction in ( w1τ 2 ) of this coefficient when calculating the other coefficients c j , M (t ) = 1 for M ≠ M 0 . 2

The present theory is easily extended to a spin S of arbitrary value. Indeed, the off-diagonal coefficients PMM 0 are still proportional to the functions P1a (t ) and P2a (t ) for M − M 0 = ±1 and

±2 , respectively, and zero otherwise, so that Eqs. (B23) and (B24) still hold. Finally, the tempting Markovian hypothesis P (tc − ta ) = P (tc − tb )P (tb − ta ) for arbitrary

times 0 ≤ ta ≤ tb ≤ tc , which would result from an incorrect application of Eq. (B5), obviously implies the validity of the usual Redfield theory and is thus incompatible with the transient behavior of the "exact" simulated TCFs at short time.




APPENDIX SB: POPULATION EVOLUTION AND TRANSITION PROBABILITIES mod (t ) for M ≠ M 0 1. Calculation of the probability PMM 0 mod The probability PMM (t ) , simply denoted by PMM 0 (t ) in the present Appendix SB and defined 0

for M ≠ M 0 by Eq. (B17)

∫

2

mod PMM (t ) = PMM 0 (t ) ≡ c j , M (t ) ≅ 0

t 0

(

)

(L) exp iω mod j , MM 0 t ' M H ZFS, j (t ') M 0 dt '

2

(B17)

is calculated as follows. The probability density to find the zM axis of the molecular frame in the direction (α 0 , β 0 ) at t = 0 is assumed to be isotropic. It is equal to 1 (4π ) . Introducing the rotational correlation times τ l of order l of the complex as τ l ≡ 1 ⎡⎣l (l + 1) D r ⎤⎦ , the rotational Brownian motion of the zM axis is described by the conditional probability density (propagator) ∞

l

ρ (α 0 , β 0 ; α , β ;τ ) = ∑ ∑ Ylm (α 0 , β 0 ) Ylm* (α , β ) exp ( −τ τ l )

(SB1)

l = 0 m =− l

to find zM in the direction (α , β ) at time t given that it was initially in the direction

(α 0 , β0 ) . The derivative dPMM dPMM 0 (t ) dt

0

(t ) dt is readily expressed in terms of ρ (α 0 , β 0 ; α , β ;τ ) as

t

= 2 Re ∫ dτ ∫ 0

(

× exp iω

mod MM 0

space α 0 , β 0

dα 0 d β 0

1 dα d β ρ (α 0 , β 0 ;α , β ;τ ) 4π ∫ space α , β

τ ) M H (τ ) M 0 M H (0) M 0 (L) 1

(L) 1

(SB2)

∗

which by straightforward integration over dα d β simplifies to

dPMM 0 (t ) dt

= 2 A2 ( S , M , M 0 ) Re ∫ dτ ∫ t

0

(

× exp iω

mod MM 0

space α 0 , β 0

dα 0 d β 0

( cos β 0 )τ − τ τ 2 ) Y2 q (α 0 , β 0 )

2

(SB3)


with A2 ( S , M , M 0 ) ≡ H ZFS


2

⎛ S ⎜ ⎝ −M

2 M − M0


2

S ⎞ mod ⎟ , q ≡ M − M 0 ≠ 0 , and ωMM 0 ( cos β 0 ) = M0 ⎠

(L) ωMM + P2 ( cos β 0 ) δωMM . For the present perturbing Hamiltonian H1(L) (t ) = H ZFS (t ) , the 0

0

parameters k MM 0 (0) and τ c to be used in Eqs. (B7) to (B13) are k MM 0 (0) = A2 ( S , M , M 0 ) and (L) τ c = τ 2 . If the modulation of ωMM by H ZFS (t ) is neglected, i.e., P2 ( cos β 0 ) δωMM τ is set to 0

0

zero, Eq. (SB3) reduces to Eq. (B12). In the general case, the probability PMM 0 (t ) is obtained from Eq. (SB3), first by integrating twice over the time variable τ , and second by integrating over dα 0 d β 0 . The two integrations over the time variable are readily performed to yield the mod auxiliary function pMM 0 (t , cos β 0 ) given by the expression of PMM 0 (t ) in Eq. (B12) with ωMM 0

replacing ωMM 0 and τ 2 replacing τ c , i.e.,

(

)

(

)

mod mod pMM 0 (t , cos β 0 ) ≡ wMM 0 ωMM ( cos β 0 ) ⎡⎣t − ζ t ,τ 2 , ωMM ( cos β 0 ) τ 2 ⎤⎦ 0 0

(SB4)

The probability PMM 0 (t ) can be rewritten as PMM 0 (t ) = ∫

pMM 0 (t , cos β 0 ) Y2 q (α 0 , β 0 ) dα 0 d β 0 2

space α 0 , β 0

= 4π ∫

π 2 0

(SB5)

pMM 0 (t , cos β 0 )Y2 q (α 0 = 0, β 0 ) d β 0 2

By changing the integration variable β 0 into x = cos β 0 , the expression (SB5) of PMM 0 (t ) further reduces to

1

PMM 0 (t ) = ∫ pMM 0 (t , x) yq ( x)dx with 0

(

)

(

)

mod mod pMM 0 (t , x) ≡ wMM 0 ωMM ( x) ⎡⎣t − ζ t ,τ 2 , ωMM ( cos β 0 ) τ 2 ⎤⎦ , 0 0 mod ( x) = ωMM 0 + P2 ( x)δωMM 0 , ωMM 0

y±1 ( x) ≡

(SB6)

2 15 2 15 x (1 − x 2 ) , y±2 ( x) ≡ (1 − x 2 ) 2 8

The expressions (SB6) of PMM 0 (t ) can be integrated numerically. They also have analytical approximations derived as follows: It was checked numerically that, if ω0 > δωMM 0 with


δωMM

0

(

defined

by

)

(


Eq.

(B16),

the

values

)


of

(

the

functions

mod wMM 0 (ωMM ), 0

)

⎡1 − ω mod τ 2 ⎤ ⎡1 + ω mod τ 2 ⎤ , and ω mod τ ⎡1 + ω mod τ 2 ⎤ involved in the expression of MM 0 2 MM 0 2 MM 0 2 ⎢ MM 0 2 ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎣ mod pMM 0 (t , x) and calculated at ωMM = ωMM 0 are good approximates to within a few percents of 0

the integrals over dx of these functions times yq ( x) . Thus, these functions are replaced by mod their values at ωMM = ωMM 0 . Under this simplifying assumption, the probability PMM 0 (t ) 0

becomes PMM 0 (t ) = wMM 0 (ωMM 0 ) ⎡⎣t − ζ qmod (t ,τ 2 , ωMM 0 , δωMM 0 )τ 2 ⎤⎦ with ⎧⎪1 − ωˆ 2 − exp ( − t τ 2 ) ⎫⎪ ζ ⎨ ⎬, 2 ⎡ ⎤ ˆ ˆ × − − 1 ω Re f ( t , ω , δω ) 2 ω Im f ( t , ω , δω ) ( ) q q ⎪⎩ ⎣ ⎦ ⎪⎭ 1 15 f ±1 (t , ω , δω ) ≡ ∫ exp [iωt + iP2 ( x) δω t ] x 2 (1 − x 2 ) dx, 0 2 1 2 15 f ±2 (t , ω , δω ) ≡ ∫ exp [iωt + iP2 ( x) δω t ] (1 − x 2 ) dx 0 8 mod q

1 (t ,τ 2 , ω , δω ) ≡ 1+ωˆ 2

(SB7)

The integrals f q (t , ω , δω ) can be computed numerically. They can also be derived from the analytical expressions given by the Mathematica™ software

∫

1 0

exp [iP2 ( x)at ]

exp ( −i at 2 ) 15 2 x (1 − x 2 ) dx = 52 2 12 ( at )

⎧ ⎡1 ⎤⎫ × ⎨−30 exp ( 3i at 2 ) at + 3π ( 5 + 5i ) (1 + iat )erf ⎢ (1 − i ) 3at ⎥ ⎬ ⎣2 ⎦⎭ ⎩

(SB8)

and

∫

1 0

exp [iP2 ( x)at ]

2 exp ( −i at 2 ) 15 1 − x 2 ) dx = ( 52 8 48 ( at )

⎧ ⎡1 ⎤⎫ × ⎨30(1 + iat ) exp ( 3i at 2 ) at + 3π ( 5 + 5i ) ⎡⎣ −1 + at ( −2i + 3at ) ⎤⎦ erf ⎢ (1 − i ) 3at ⎥ ⎬ ⎣2 ⎦⎭ ⎩

(SB9)




2. The negligible influence of the rhombicicity of the ZFS Hamiltonian

FIG. S3. S = 3/2, ∆S = 1 cm-1, B0 = 7.05 T. Short-time decay of the simulated TCFs g kk00 (t ) for a static ZFS Hamiltonian of magnitude ∆S = 1 cm-1 having the

(

)

largest possible rhombicity given by the parameters DS = 3 2 4 ∆S , ES = DS 3 and fluctuating with a rotational correlation time τ 2 = 100 ps. For comparison, the functions g kk00,mod (t ) derived from the evolution Eq. (B23) of the theory of ZFSmodulated transition frequencies are also plotted in the case a static axial ZFS Hamiltonian having the same magnitude ∆S = 1 cm-1 with the parameters DS = 3 2∆S = 1 cm–1, ES = 0 and the same rotational dynamics.




APPENDIX SC: POPULATION EVOLUTION AND REDFIELD THEORY

To be complete, it will be reminded that within the Redfield limit the population evolution equation dσ diag dt = Wσ diag is formally identical to the Redfield master equation.7 Defining the statistical operator σ I (t ) in the interaction picture as

σ I (t ) = eiH tσ (t )e −iH t 0

(C1)

0

the Redfield relaxation formalism reads I dσ MM ' (t ) = ∑ RMM ', M 0 M '0 (ω0 )σ MI 0 M '0 (t ) dt M 0 , M '0

(C2)

where the sum is restricted by the condition M − M ' = M 0 − M '0 and the Redfield matrix elements RMM ', M 0 M '0 (ω0 ) are given by Eq. (5) of Ref.6. The system of equations (C2) splits into I independent sub-systems of equations indexed by M − M ' = q . Since σ MM (t ) = σ MM (t ) , the

sub-system corresponding to q = 0 reduces to the Redfield master equation S dσ MM (t ) = ∑ AMM 0 σ M 0 M 0 (t ) dt M 0 =− S

(C3)

AMM 0 = RMM , M 0 M 0 (ω0 )

(C4)

with

From Eq. (5) of Ref.6, we have for M ≠ M 0 ∞

∗

(L) (L) AMM 0 = AM 0 M = 2 Re ∫ exp(iωMM 0τ ) M H ZFS (t ) M 0 M H ZFS (t − τ ) M 0 dτ 0

and

(C5)


J. Chem. Phys. (2007) AMM = −

∑

M0 ≠M

AMM 0

Supplementary Material S24 (C6)

According to Eq. (21), the expression (22) of wMM 0 is identical to that of AMM 0 given by Eq. (C5) for M ≠ M 0 and Eq. (C6) for M = M 0 , so that we have wMM 0 = AMM 0

(C7)

The population evolution equation dσ diag dt = Wσ diag is identical to the Redfield master equation (C3).




APPENDIX SD: REDFIELD RELAXATION MATRICES

The analytical expressions of the Redfield TCFs g kkqq' (t ) are obtained for each q value from the time-evolution of the

n = 2S + 1 − q

coefficients

X M( q ) (t ) ≡ σ M + q , M (t )

( M ≤S,

M + q ≤ S ) of the band q of the density matrix. According to Eq. (6) of Ref.6 the variation

rates of these coefficients obey the Redfield linear differential equation

d (q) (q) (q) X M = ∑ RMM 'XM' dt M'

(SD1)

(q ) (q) where the elements RMM of order q are the ' of the Redfield relaxation sub-matrix R

6 (q) elements RMM ' ≡ RM + q , M ; M ' + q , M ' of the full Redfield relaxation matrix given by Eq. (5) of Ref. .

The coefficients X M( q ) (t ) at time t are given in terms of their initial values by the exponential relation

(X

(q) M

(t ) ) = exp ⎡⎣ R ( q )t ⎤⎦ ( X M( q ) (0) )

(SD2)

Let − Re,λ + ise,λ ( 1 ≤ λ ≤ n ) be the eigenvalues of R ( q ) and u λ ≡ ( uM λ ) the associated eigenvectors in the basis M + q M

of the subspace of the (2S + 1) × (2S + 1) matrices

spanned by the band matrices of index q . The quantities Re,λ with Re,λ ≥ 0 define the electronic (e) decay rates. The image of

uλ

by the exponential of

R ( q )t

is

exp ⎡⎣ R ( q )t ⎤⎦ u λ = exp ⎡⎣( − Re,λ + ise,λ ) t ⎤⎦ u λ . Denote the coefficients of the operators tkq and tkq' in the basis u λ by cλ and c 'λ so that tkq = ∑ cλ u λ and tkq' = ∑ c 'λ u λ . Then, the TCF g kkqq' (t ) is λ

simply given by

λ



{

g kkqq' (t ) = exp ( iqω0t ) tr tkq † exp ⎡⎣ R ( q )t ⎤⎦ tkq'


}

† ⎧⎪⎛ ⎞ ⎛ ⎞ ⎫⎪ = exp ( iqω0t ) tr ⎨⎜ ∑ cλ u λ ⎟ exp ⎡⎣ R ( q )t ⎤⎦ ⎜ ∑ c 'λ ' u λ ' ⎟ ⎬ ⎠ ⎝ λ' ⎠ ⎭⎪ ⎩⎪⎝ λ

(SD3)

= exp ( iqω0t ) ∑ c 'λ ' exp ⎡⎣( − Re,λ ' + ise,λ ' ) t ⎤⎦ ∑ cλ* ∑ uM* λ uM λ ' λ'

λ

M

In the simple case where the eigenvectors u λ form an orthonormal basis, the expression (SD3) of g kkqq' (t ) simplifies to

g kkqq' (t ) = exp ( iqω0t ) ∑ c 'λ cλ* exp ⎡⎣( − Re,λ + ise,λ ) t ⎤⎦

(SD4)

λ

For S = 1 and S = 3 2 , the various relaxation matrices R ( q ) ( q ≥ 0 ) together with their eigenvalues and eigenvectors, which are used to derive the Redfield TCFs compared with the simulation results, are listed herafter.

1. Spin S = 1 q = 0:

W=R

(0)

2 j (2ω0 ) j (ω0 ) ⎛ − j (ω0 ) − 2 j (2ω0 ) ⎞ 1⎜ ⎟ = ⎜ −2 j (ω0 ) j (ω0 ) j (ω0 ) ⎟ real symmetric 5⎜ 2 j (2ω0 ) j (ω0 ) − j (ω0 ) − 2 j (2ω0 ) ⎟⎠ ⎝

eigenvalues: − Re,0 = 0 , − Re,1 = −

1 3 [ j (ω0 ) + 4 j (2ω0 )] , − Re,2 = − j (ω0 ) 5 5

⎛ −1⎞ ⎛ 1⎞ ⎛1⎞ 1 ⎜ ⎟ 1 ⎜ ⎟ 1 ⎜ ⎟ orthonormal eigenvectors: u 0 = −2 1 , u1 = 0 , u2 = 3 ⎜⎜ ⎟⎟ 2 ⎜⎜ ⎟⎟ 6 ⎜⎜ ⎟⎟ ⎝1⎠ ⎝1⎠ ⎝ 1⎠

tk0 expansion in the basis u λ : t00 = u 0 , t10 = u1 , t20 = u 2




q =1:

R (1)

1 ⎛ ⎞ (1) − j (ω0 ) ⎟ ⎜ R−1,−1 5 =⎜ ⎟ (1) ⎜ − 1 j (ω ) R0,0 ⎟⎟ ⎜ 0 ⎝ 5 ⎠

(1) =− with R−(1)1,−1 = R0,0

1 ⎡3 j (0) + 3 j (ω0 ) + 2 j (2ω0 ) + iω0τ 2 ( j (ω0 ) + 4 j (2ω0 ) ) ⎤⎦ 10 ⎣

eigenvalues:

− Re,1 = −

1 ⎡3 j (0) + 5 j (ω0 ) + 2 j (2ω0 ) + iω0τ 2 ( j (ω0 ) + 4 j (2ω0 ) ) ⎤⎦ , 10 ⎣

− Re,2 = −

1 ⎡3 j (0) + j (ω0 ) + 2 j (2ω0 ) + iω0τ 2 ( j (ω0 ) + 4 j (2ω0 ) ) ⎤⎦ 10 ⎣

orthonormal eigenvectors: u1 = −

1 ⎛ 1⎞ 1 ⎛1⎞ ⎜ ⎟ , u2 = ⎜ ⎟ 2 ⎝ 1⎠ 2 ⎝ −1⎠

tk1 expansion in the basis u λ : t11 = u1 , t21 = u 2

q = 2: R (2) = −

1 ( j (ω0 ) + 2 j (2ω0 ) + iω0τ 2 ( j (ω0 ) + 4 j (2ω0 ) ) ) 5

eigenvalue: 1 − Re,2 = − ⎡⎣ j (ω0 ) + 2 j (2ω0 ) + iω0τ 2 ( j (ω0 ) + 4 j (2ω0 ) ) ⎤⎦ 5 orthonormal eigenvector: u 2 = (1)

t22 expansion in the basis u λ : t22 = u 2




2. Spin S = 3/2

q = 0:

W = R (0)

with w1 ≡

⎛ − w1 − w2 ⎜ w1 ⎜ = ⎜ w2 ⎜ 0 ⎝

w1 − w1 − w2

w2 0

0

− w1 − w2

w2

w1

⎞ ⎟ ⎟ real symmetric w1 ⎟ ⎟ − w1 − w2 ⎠ 0 w2

6 6 j (ω0 ) , w2 ≡ j (2ω0 ) 5 5

eigenvalues: − Re,0 = 0 , − Re,1 = −

12 12 12 j (ω0 ) , − Re,2 = − [ j (ω0 ) + j (2ω0 ) ] , − Re,3 = − j (2ω0 ) 5 5 5

⎛ −1 ⎞ ⎛1⎞ ⎛1⎞ ⎛ −1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 1 1 1 1 −1 1 −1 orthonormal eigenvectors: u 0 = ⎜ ⎟ , u1 = ⎜ ⎟ , u 2 = ⎜ ⎟ , u3 = ⎜ ⎟ 2 ⎜1⎟ 2 ⎜ −1 ⎟ 2 ⎜ −1⎟ 2⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝1⎠ ⎝1⎠ ⎝1⎠ ⎝1⎠

tk0 expansion in the basis u λ : t00 = u 0 , t10 =

1 2 2 1 u1 + u3 , t20 = u 2 , t30 = u1 − u3 5 5 5 5

q = 3: R (3) = −

6 ( j (ω0 ) + j (2ω0 ) + iω0τ 2 ( j (ω0 ) + 2 j (2ω0 ) ) ) 5

eigenvalue: 6 − Re,3 = − ⎡⎣ j (ω0 ) + j (2ω0 ) + iω0τ 2 ( j (ω0 ) + 2 j (2ω0 ) ) ⎤⎦ 5 orthonormal eigenvector: u3 = (1)

t33 expansion in the basis u λ : t33 = u3




Time-shifted Redfield and McLachlan approximations of g1100 (t ) in the presence of fluctuating static and transient ZFS Hamiltonians

Time-shifted Redfield approximation:

Eq. (34) transforms into 1 4 (1) (1) (2) (2) − (t + τ v ) T1e,T + exp ( − (t + τ 2 ) T1e,S − (t + τ v ) T1e,T g1100 (t ) = exp ( − (t + τ 2 ) T1e,S ) ) (SD5) 5 5

Time-shifted McLachlan approximation:

Eq. (39) transforms into McLachlan McLachlan g1100,t-shift McL (t ) = exp ( − (t + τ 2 ) T1e,S − (t + τ v ) T1e,T )

(SD6)




The negligible influence of the rhombicicity of the ZFS Hamiltonian:

FIG. S4. S = 3/2, B0 = 7.05 T. The Monte-Carlo (MC) simulated TCF g1100 (t ) compared with the time-shifted biexponential Redfield TCF derived from Eq. (34) for a static ZFS Hamiltonian of magnitude ∆S = 1 cm-1 having the largest possible

(

)

rhombicity given by the parameters DS = 3 2 4 ∆S , ES = DS 3 and fluctuating with a rotational correlation time τ 2 = 100 ps.. The time-shifted McLachlan approximation derived from Eq. (39) is less accurate.

REFERENCES OF THE SUPPLEMENTARY MATERIAL 1

A. Messiah, Mécanique Quantique. (Dunod, Paris, 1972).

2

J. Kowalewski and L. Mäler, Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications. (Taylor & Francis, London, 2006).

3

A. Abragam, Les Principes du Magnétisme Nucléaire. (PUF, Paris, 1961).

4

C. P. Slichter, Principles of Magnetic Resonance. (Springer-Verlag, Berlin, 1978).

5

Y. Ayant and E. Belorizky, Cours de Mécanique Quantique. (Dunod, Paris, 1969).

6

S. Rast, P. H. Fries, and E. Belorizky, J. Chem. Phys. 113, 8724 (2000).

7

A. G. Redfield, Adv. Magn. Reson. 1, 1 (1965).