Inter system crossing

Why the need for "inter system crossing" and what it is

The need for flipping spins to make some covalent reactions happen

In piezochemical mechanosynthesis sometimes one wants to intentionally flip a spin.
Without the spin being flipped the lowest possible electronic energy state (in the timeframe of the mechanosynthetisation reaction)
might not be the ground state. That means when pressing the bonding partners together they might not form a covalently bonding orbital.
This is due to the pauli-principle not allowint two electrons with the same spin sitting in the same (lowest energy state) orbital.

Spin-flipping less critical for heavier atoms like transition metals:
This is especially the case with light elements like carbon where filled high-energy antibinding orbitals often make a bond impossible.
Transition metals can from bonds with electrons in antibinding orbitals (albeit weaker bonds) and these eventually de-excite to become bonds of full strength.
Still this is not at all desirable since this is a high energy dissipation mechanism. Devaluating a lot of un-bond Helmholtz free energy.

Flipping a spin means flipping (preserverd) angular momentum => drain needed

Since

• spin is linked to angular momentum (gyromagnetic factor) and
• there is conservation of angular momentum

we need to find a recipient for that angular momentum that is both

• capable of taking that angular momentum up quickly and
• "willing" to take it up quickly.

Listing potential means for flipping spins

What we seek are effects that make some angular momentumm carrying "particles" interact with electron spins in the form of a magnetic torque.
Options for potential sinks for unwanted angular momentum that comes with spin include:

• (?photons in the form of phosphorescence? – way too slow end dissipating energy)
• nuclear spins (energy conservation too)
• external magnetic field
• orbital configuration change – via "spin orbit coupling"

It seems the last one is best (to check what's the issue with the others)

Why spins are "frobidden" to flip without relativistic quantum mechanics

Unfortunately paired spins (singlet-state wave function) and unpaired spins (triplet-state wave function) are orthogonal wave functions
meaning their overlap integral is zero – meaning they don't interact – meaning the transition is "forbidden".
(Meaning Fermis golden rule predict infinitely long transition time?)
So there's a problem with the "willingness" part.
This is just to first approximation though.

Relativistic quantum mechanics saving the day

Taking into account effects of relativistic quantum mechanics,
(these fall out of the Dirac equation – the relativistic generalization of the Schrödinger equation)
One gets a spin-orbit coupling energy term in the Hamiltonian. Exactly what we want.
In the non-relativistic picture this effect can be fudged in as a perturbation term. Spin orbit coupling term (SO).

[math] H_{SO} = - \frac{Z^4 e^2}{8 \pi \epsilon_0 m_e^2 c^2} \vec l \cdot \vec s [/math]

Note that the interaction strength is proportional to the fourth power of the atomic number Z.
Doubling the atomic number causes a 16-fold increase in interaction strength.
So the Elements of the 3rd row and higher are good "drains" for getting rid of unwanted angular momentum.

• This is called the "heavy atom effect".
  
• It even works if there is no direct covalent bond "external heavy atom effect"
  

How does the latter one work? There needs to be some overlap of electron wave functions right?
A highly electronically isolating connection (like plain diamond can be, or a nonbonding VdW contact can be, or a minimal vacuum gap can be) is probably a bad idea right?

Side-notes to spinning flips via ISC

• Beside "spin flipping" there is also "spin rephaseing" as a possible transition (precession speed ...) ???
• This topic is related to spectroscopy and optical effects

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Inter system crossing in (piezochemical) mechanosynthesis

Design target: Increasing inter system crossing rates. Goals:

• avoiding errors
• increasing reaction speeds
• reducing energy dissipation

Avoiding errors (omitted reactions, misreactions) due to too low singlet-triplet energy gap

• Singlet transition geometries often resemble triplet equilibrium geometries (Salem and Rowland 1972 – referenced by Nanosystems)
• Singlet state (paired) to low lying triplet state is undesired

To avoid that

• either (conservatively) get the energy gap up to above 145zJ ( = k * 300K * ln(10^15) )
• or (weaker prerequisite) get the accumulated ramp-on ISC transition rate to a probability above ln(P_err) prior
  to theΔV_s,t point where the geometry is no longer suitable for bond formation

Inadvertently slowing down by pressing to hard

Citations from Nanosystems:

• "As the radicals approach the gap between the triplet and singlet state energies grows, but this decreases the rate of intersystem crossing".
  
• "The condition that ΔV_s,t ≥145maJ imposes a significant constraint because k_isc varies inversely with the electronic energy difference ΔV_isc which (in the absence of mechanical relaxation) would equal the difference in equilibrium energies ΔV_s,t, and will frequently be of similar magnitude"

Avoiding dissipation during bond cleavage by speeding up ISC

If inter system crossing is slow compared to (tensile) bond cleavage then cleavage gives a singlet diradical
When pulling the cleaved parts further apart then the singlet triplet energy gap vanishes
and thermal excitations fill the triplet state (a loss in Helmholtz free energy of minus ln(2)kT – a loss of one bit information – undesired dissipation)

(TODO: That would cause local cooling that goes unused and thus gets dissipated by ambient heat flowing in? Could that be partially recuperated by a heat pump? Abd or exploited for deliberate cooling?)

If inter system crossing is fast compared to (tensile) bond cleavage then

• thermal excitations fill the (repulsive) triplet state already during bond cleavage
• => there is a reduction of the mean-force bond potential energy (?)
• there is no significant dissipation ☺

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An other means for reducing dissipation is: Dissipation sharing

ISC rates in pi-bond twisting

Abstraction of a moiety to yield an aklene (accelerating Diels Adler and related reactions)

• resembles radical coupling
• requires spin pairing
• raises questions about inter-system crossing rates

Employing the "external heavy atom effect" to accelerate ISC

Nearby site integration of heavy elements.
E.g. Bismuth (Z=83) – since it likes to form 3 weak covalent bonds (?) (suggested in Nanosystems)

Given known examples for the "external heavy atom effect" (Nanosystems page 216) it should be possible to have:

• k_isc>10^9 with ΔV_s,t >145zJ and thus
• t_trans<10^-7s with P_err<10^-15

Nanosystems references

• 8.3.4. Preview: molecular manufacturing and reliability constraints – e. Meeting constraints on omitted reactions in a single trial. (P210 center)
• 8.4.4. Carbon radicals – b. Radical coupling and inter system crossing (P215 bottom, P216)
• 8.5.3. Tensile bond cleavage – c. Spin, dissipation, and reversibility. (P224 bottom)
• 8.5.6. Pi bond torsion (P231 bottom)

Related

• Piezochemical mechanosynthesis
• Mechanosynthesis
• Fun with spins – influencing spins by ligand-fields (crystal-fields) rather than spin-orbit coupling
• Quantum mechanics
• Photonics
• Electronic transitions

External links

(Wikipedia: Intersystem crossing)

• Source of info in the intro: Video: [1]