Scheepers, ranked 42nd in the world, has seen her summer hardcourt form dip after playing a Stanford quarter-final three weeks ago against Serena Williams.
After losing 6-4, 6-0 to Williams, the 28-year-old Scheepers won a round the next week in Carlsbad before losing to US Olympian Varvara Lepchenko.
The second seed never really got started in the heat and humidity of the US capital as she lost serve five times in 65 minutes to Rybarikova, the world number 114 who won despite making the long haul from the WTA event in Baku, central Asia.
Romanian Edina Gallovits-Hall upstaged Czech fifth seed Barbora Zahlavova Strycova 7-5, 4-6, 6-3.
Scheepers and Zahlavova's losses were counter-balanced by two seeded wins.
Eighth seed Olga Govortsova of Belarus beat Karolina Pliskova of the Czech Republic 6-2, 6-2 and American seventh seed Coco Vandeweghe defeated Japan's Erika Sema 6-4, 6-1.
Vandeweghe reached the Stanford final as a qualifier, losing to Williams in the title match, but running her WTA ranking up to its current 70th. She fought off disappointment the week following her breakthrough by losing in the Carlsbad first round to Scheepers only two days after facing Williams.
The Washington tournament is being played as a joint event for the first time under the wing of a new sponsor.
But the London Olympic mens and ladies tournament has drained the talent pool, with the highest seed on the women's side being Russian world number 28 Anastasia Pavlyuchenkova. On the men's side, there are just two players ranked in the top 30 in the world.
The men's draw is headed by Mardy Fish, who decided months ago - even before an accelerated heartbeat gave him a now-cured health scare - to skip the Games.
The world number 13 who has played a limited schedule this season and missed two months with his medical condition. He starts play on Tuesday against German Bjorn Phau.
That pair have played three times, always in the US, with the 30-year-old Fish standing 2-1 in the series. Fish, whose best showings were a pair of quarter-finals in 2003 and 2006 from six previous appearances.
Ukrainian Alexandr Dolgopolov is making his tournament debut as second seed, ahead of South African Kevin Anderson.
In the men's first round, Los Angeles quarter-finalist Leonardo Mayer of Argentina beat Spaniard Guillermo Garcia-Lopez 7-6 (7/5), 7-6 (10/8), a battle lasting just under two hours.
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