md2011-04-note_Part_7

md2011-04-note_Part_7 - U LJ ( r ) = 4 r 12- r 6 . (29) I...

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I However, ab initio calculations give [Solid State Physics: Advances in Research and Applications, 43 (1990) 1] : Element - E coh (eV) E f (eV) V 5.31 2.1 ± 0.2 Nb 7.57 2.6 ± 0.3 W 8.90 4.0 ± 0.2 I This comparison neglects the effect of relaxation, but in simple metals it’s not likely to have an effect larger than 1 eV. I Better, and more computationally expensive, potentials will be presented during the coming lectures. Notes
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P±²³´²µ¶·¸ ¶² C¹²±ºº I Since we have to limit the interaction to a certain distance r c from each atom, this means that we have a discontinuity in the U ( r ) . I Discontinuity in the potential leads to a jump in the force at r c , which makes the physics of a simulation questionable. U ( r ) r r cut I So, we need to have a potential which has a continuous first derivative. I To achieve this at r c , we must smoothly drive the potential to zero within a certain distance range r [ r c , r c + Δ r ] . I In many modern potentials, the potential is defined with a proper first derivative at r c . Notes
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Driving a Lennard-Jones potential to zero I As we saw, Lennard-Jones is defined as U ( r ) r Pauli repulsion Equilibrium Dipole-dipole interaction
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Unformatted text preview: U LJ ( r ) = 4 r 12- r 6 . (29) I There are (at least) two ways how to drive this potential to zero: I Shift and tilt the potential to get U ( r ) and U ( r ) continuous at r c . I Use a third order polynomial for r [ r c , r c + r ] . Notes I Shift-and-tilt requires playing around with the actual potential equation: U ( r ) = U LJ ( r ) - ( r-r c ) U LJ ( r c ) -U LJ ( r c ) . (30) I Now U ( r c ) = 0 as is U ( r c ) = 0. I However, since we are changing the equation, the tting must be redone with the new equation. U ( r ) r shift-and-tilt U LJ P ( r ) I Or, as said, we can solve constants for the polynomial P ( r ) = ar 3 + br 3 + cr + d for r [ r c , r c + r ] with conditions: P ( r c ) = U LJ ( r c ) P ( r c ) = U LJ ( r c ) P ( r c + r c ) = P ( r c + r c ) = (31) Notes I Note that even smooth-looking potential functions can have problematic forces (even if continuous). Notes...
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This note was uploaded on 02/14/2012 for the course CSE 6590 taught by Professor Kotakoski during the Winter '12 term at York University.

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md2011-04-note_Part_7 - U LJ ( r ) = 4 r 12- r 6 . (29) I...

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