GOV SMT TAXIED INTO POSITION ON WRONG RWY CAUSING NEAR COLLISION WITH SMT ON APCH IN REDUCED VISIBILITY AT RIC. SMT ON APCH LANDED OVER TOP OF SMT IN POSITION.

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I RECEIVED THE ATIS AND OUR CLRNC TO IAD. I CONTACTED GND CTL AND RECEIVED CLRNC TO TAXI TO RWY 16; HOLD SHORT OF RWY 20. THE CHKLIST WAS CALLED OUT AND COMPLETED WHILE HOLDING SHORT OF RWY 20. I THEN CONTACTED THE TWR AND TOLD THEM WE WERE READY TO GO IN SEQUENCE. I UNDERSTOOD THAT WE WERE CLRED INTO POS AND HOLD ON RWY 16. I ACKNOWLEDGED AND READ BACK THAT WE WERE XING RWY 20 TO HOLD ON RWY 16. WHILE IN POS ON RWY 16 WE COMPLETED THE FINAL ITEMS ON THE TKOF CHKLIST AND I TURNED ON THE RADAR TO SEE IF I COULD GET A POS ON WX JUST S OF THE AIRFIELD. I MADE A REMARK TO THE OTHER PLT THAT WE NEEDED TO TURN LEFT AS SOON AS POSSIBLE AFTER TKOF. HE REPLIED AGREEMENT AND SAID I WISH THEY WOULD HURRY AND GIVE US TKOF CLRNC. WE MONITORED A XMISSION FROM THE TWR TO A LNDG ACFT ABOUT THE VISIBILITY BEING 1 MI AND THERE SHOULD NOT BE A PROB LNDG. WE ASSUMED THIS ACFT WAS LNDG ON RWY 20. THE NEXT XMISSION WAS FROM AND ACFT SAYING: 'MY GOD; THERE IS AN ACFT ON THE RWY.' THIS ACFT LANDED OVER US. THE TWR THEN ASKED US WHAT RWY WERE WE ON? I RESPONDED; '16.' THE TWR SAID YOU WERE CLRED ONTO RWY 20 AND TO IMMEDIATELY CLR THE RWY. WE TAXIED CLR OF THE RWY AND WERE THEN GIVEN INSTRUCTIONS TO TKOF ON RWY 20. ON THIS PARTICULAR MORNING THERE MAY HAVE BEEN A FALSE SENSE OF SECURITY ON EVERYONE'S PART. GENERALLY; WHEN IN S OPERATION; ACFT DEPARTING RICHMOND TO THE N CAN EXPECT TO BE ASSIGNED RWY 16; THEN A LEFT TURN TO THE E AND THEN ANOTHER LEFT TURN CLBING ON COURSE. ALSO THE APCHING RAIN SHOWERS FROM THE S WAS A CONCERN FOR ALL. IN SUMMARY; THERE WAS NEVER ANY DOUBT IN MY MIND THAT WE HAD BEEN CLRED INTO POS ON RWY 16.